Bulletin of Mathematical Biology

, Volume 76, Issue 7, pp 1809–1834 | Cite as

Modeling and Analysis of a Density-Dependent Stochastic Integral Projection Model for a Disturbance Specialist Plant and Its Seed Bank

  • Eric Alan Eager
  • Richard Rebarber
  • Brigitte Tenhumberg
Original Article


In many plant species dormant seeds can persist in the soil for one to several years. The formation of these seed banks is especially important for disturbance specialist plants, as seeds of these species germinate only in disturbed soil. Seed movement caused by disturbances affects the survival and germination probability of seeds in the seed bank, which subsequently affect population dynamics. In this paper, we develop a stochastic integral projection model for a general disturbance specialist plant-seed bank population that takes into account both the frequency and intensity of random disturbances, as well as vertical seed movement and density-dependent seedling establishment. We show that the probability measures associated with the plant-seed bank population converge weakly to a unique measure, independent of initial population. We also show that the population either persists with probability one or goes extinct with probability one, and provides a sharp criteria for this dichotomy. We apply our results to an example motivated by wild sunflower (Helianthus annuus) populations, and explore how the presence or absence of a “storage effect” impacts how a population responds to different disturbance scenarios.


Disturbance specialist Seed bank Integral Projection Model Weak convergence Density dependence Storage effect 

1 Introduction

In many plant species dormant seeds persist in the soil from one to several years (Roberts 1981; MacDonald and Watkinson 1981; McGraw 1986; Doyle et al. 1986; Maxwell et al. 1988; Venable 1989; Doyle 1991; Kalisz and MA, 1992; Jordan et al. 1995; Gonzalez-Andujar 1997; Cummings et al. 1999; Alexander et al. 2001; Cummings and Alexander 2002; Edelstein-Keshet 2005; Fenner and Thompson 2005; Pekrun et al. 2005; Colbach et al. 2008; Garnier et al. 2006 ). By allowing individuals to disperse through time, these seed banks buffer against the effects of environmental variation on population size and thus enhance persistence. Such environmental buffering is especially important for annual plant species, which can only reproduce once. Many annual plants are disturbance specialists, germinating only in disturbed soil (Alexander and Schrag 2003). Disturbances can create a more favorable environment for germination and recruitment by removing more competitive species and enhancing seed bank formulation. In these species factors such as the frequency, intensity, timing, and spatial extent of disturbances can greatly influence germination and survival rates of seeds in the seed bank (Froud-Williams et al. 1984; Claessen et al. 2005a; Moody-Weis and Alexander 2007; Miller et al. 2012).

Disturbances alter the depth distribution of seeds in the seed bank, burying some seeds deep in the soil where survival is high (and germination rates are low), and relocating other seeds closer to the soil surface where germination rates are high (but survival is low) (Moody-Weis and Alexander 2007; and reviewed in Mohler 1993). This has attracted much attention in agricultural research, and many models have been constructed to explore different ploughing regimes to manipulate the distribution of weed seeds in the soil in order to reduce weed population size in agriculture (Doyle et al. 1986; McGraw 1986; Maxwell et al. 1988; Mohler 1993; Jordan et al. 1995; Gonzalez-Andujar 1997; Mertens et al. 2002; Pekrun et al. 2005; Garnier and Lecomte 2006; Sester et al. 2007; Berg et al. 2010). However, in addition to anthropogenic disturbances, many natural processes move seeds to different soil depths. For example, seeds can be moved from the surface to lower soil depth through earthworm cast and mole burials. Seeds can be moved from lower seed depths to the surface through settling of the soil or digging activities by mammals, to name a few instances. In this manuscript, we consider the population dynamics of disturbance specialist plants in a natural environment, where disturbances occur in a more unpredictable fashion than those that occur in agriculture. To the best of our knowledge, only three papers (Claessen et al. 2005a, b; Eager et al. 2013a) have considered disturbances in natural environments. In contrast to this manuscript and Eager et al. (2013a), Claessen et al. (2005a) and Claessen et al. (2005b) do not consider disturbance specialist plants.

To study the population dynamics of a general disturbance specialist, we construct a density-dependent stochastic integral projection model (IPM—Ellner and Rees 2006; Ellner and Guckenheimer 2006; Ellner and Rees 2007). In contrast to traditional stochastic matrix models, where stage variables are discrete (see, for example, Tuljapurkar 1990; Caswell 2001), IPMs incorporate continuous stage variables. In our model seed depth in the soil is a continuous stage variable, and germination is only possible in the presence of a disturbance. We incorporate the characteristics of an environmental disturbance as a stochastic process \(\{\omega _t\}_{t=0}^{\infty }\) of random variables that are roughly independent and identically distributed (iid) in a probability space \(\varOmega \) of environmental conditions, with our definition of “roughly iid” below. While many random processes in biology are correlated in time (Heino et al. 2000; Vasseur and Yodzis 2004; Lögdberg and Wennergren 2012; Mustin et al. 2013), for many disturbance specialist plants (e.g. wild sunflower Helianthus annuus) the assumption of roughly iid environmental conditions is a reasonable one (Diana Pilson personal communication). In most (defined below) time-steps, we assume that there is a probability \(h\) that the population is disturbed and the depth of disturbance is distributed via a probability density function \(\rho _2(\cdot )\) with \(\text {supp}(\rho _2) = [\sigma ,D]\), where \(\sigma \ll 1\). We further assume that the disturbance in each time-step uniformly redistributes all seeds between the surface of the soil and the disturbance depth \(\omega _2\), and that the seeds below depth \(\omega _2\) remained roughly in place. We will perform a dimensional analysis on our model to ensure that the assumptions used in the model construction are consistent dimensionally, which is often ignored in ecological models (see, for example, Eager et al. 2012).

Many stochastic models for biological populations, including the one in this manuscript, can be analyzed mathematically by writing the population projection model as a difference equation of the form
$$\begin{aligned} x_{t+1} = H(\omega _t,x_t), \; \; t = 0,1,2,\ldots \end{aligned}$$
where \(x_t\) is some measurement of the population during the tth time-step, which is in a state space \(X\) and \(\{\omega _t\}_{t=0}^{\infty }\) is a sequence of random variables from a probability space of environmental conditions \(\varOmega \). The operators \(H(\omega , x)\) can be nonlinear, which is needed in order to realistically model population dynamics subject to density-dependent feedbacks. Systems of this type have been studied under many biologically motivated assumptions. For example, when \(x_t\) is positive real, consider \(H\) which satisfies the following conditions: for each \(\omega \in \varOmega \), \(H(\omega ,\cdot )\) is positive, increasing, and bounded; \({\displaystyle H(\omega ,x)/x}\) is decreasing as a function of \(x\); and \(\{\omega _t\}_{t=0}^{\infty }\) is a sequence of iid random variables. In this case, the asymptotic properties of \(x_t\) have been resolved by Ellner (1984). Similarly, when \(x_t\) is a positive vector in \({\mathbb {R}}^n\), consider \(H\) which satisfies the following conditions: for each \(\omega \in \varOmega \), the map \(H(\omega ,x) := A(\omega ,x)x\) (where \(A(\omega , x)\) is a matrix) is monotone, non-negative, and primitive; \({\partial A(\omega ,x)_{ij}}/{\partial x_l} \le 0\) for every matrix/vector entry \((i,j)\); and \(\{\omega _t\}_{t=0}^{\infty }\) is an ergodic stationary sequence. In this case, the asymptotic properties of \(x_t\) have been resolved in Benaïm and Schreiber (2009) and Schreiber (2012). When \(x_t\) is a continuous function defined on a compact set in \(\mathbb {R}^n\), consider systems for which: the operators \(H(\omega ,\cdot )\) are positive and continuous; similar conditions to those for the matrix models in Benaïm and Schreiber (2009) hold; and \(\{\omega _t\}_{t=0}^{\infty }\) is an iid sequence. In this case, the asymptotic properties of \(x_t\) have been resolved in Hardin et al. (1988). In the current paper, we extend the results of Hardin et al. (1988) so that they apply to a density-dependent stochastic integral projection model for the coupled plant-seed bank dynamics of a general disturbance specialist plant population in a random environment. The techniques we develop will allow the results in Hardin et al. (1988) to be portable to a range of systems where disturbance drives the dynamics of the population.

We prove that the sequence of probability measures \(\{\mu _t\}_{t = 0}^{\infty }\) associated with the plant-seed bank population \(\{x_t\}_{t=0}^{\infty }\) converges weakly to a stationary measure, independent of the initial population, i.e., the population \(x_t\) converges to a stationary distribution of populations. To prove this result, the techniques of Hardin et al. (1988) need to be modified somewhat. For instance, the models in Hardin et al. (1988) deal with the stochastic population dynamics of a single population over a rectangular spatial domain, while the model in this paper deal with seed bank populations structured by vertical depth, which is represented by an interval \([0,D]\), and is coupled with a scalar plant population. Perhaps more significantly, the results in Hardin et al. (1988) were probably not intended for the analysis of disturbance specialists. This is because the plant-seed bank population in our model, when viewed as a coupled system in \(C[0,D] \otimes \mathbb {R}\), is repeatedly on the boundary of the positive cone of \(C[0,D] \otimes \mathbb {R}\), since a disturbance specialist plant’s seeds cannot germinate without a disturbance (and thus the plant population is zero in years without a disturbance) and the results in Hardin et al. (1988) do not apply to systems with this behavior.

Through the intuition gained via our mathematical modeling process, we are able to work around the mathematical difficulty of the population repeatedly being on the boundary of the positive cone of \(C[0,D] \otimes \mathbb {R}\) by combining the plant and seed bank populations into one population and restricting our attention to a subset of the time steps during which the population avoids the boundary. We show that there is a dichotomy of long-term behavior of the population dynamics. As \(t \rightarrow \infty \) the population either persists with probability \(1\) (is stochastically bounded from below, Chession 1982; Ellner 1984; Benaïm and Schreiber 2009; Schreiber 2012) or goes extinct with probability \(1\), and this outcome depends on the likelihood of population growth when the population is rare, which is of importance in ecological models (Gillespie 1973; Turelli 1978; Chesson and Warner 1981; Hardin et al. 1988; Caswell 2001; Benaïm and Schreiber 2009; Schreiber 2012). This is an interesting theoretical result, since for many parameter combinations it is observed in finite-time simulation experiments that plant-seed bank models for disturbance specialists elicit boom or bust dynamics (Eager et al. 2013a), with nothing resembling a stochastic lower bound. The theory provided in this manuscript would, therefore, suggest that parameter combinations that elicit boom or bust in that paper eventually lead to asymptotic population extinction.

To conclude the paper, we pose some possible extensions to our model and discuss the mathematical properties of these models. We also provide some simulation results for an example motivated by wild sunflower (Helianthus annuus) populations that show how the disturbance parameters affect the long-term size of the population and the ability of the population to persist as \(t \rightarrow \infty \). We show that the presence or absence of a “storage effect” in the seed bank can change the way the long-term population responds to the changes in overall disturbance regimes.

2 The Model

The model in this paper is a slightly adjusted version of the model in Eager et al. (2013a), where the seed bank population evolved in the function space \(L^{1}[0,D]\). The seed bank population in this paper will end up evolving in \(C[0,D]\), which, at this time, is essential for analytic tractability. However, the qualitative results of the numerical explorations in this paper are the same as those in Eager et al. (2013a), suggesting that this modification is justifiable.

Let the positive cone of \(C[0,D]\) be
$$\begin{aligned} C[0,D]^{+} = \{f \in C[0,D] \mid f(x) \ge 0 \,\,\, \text {for} \,\,\, \text {all} \,\,\, x \in [0,D]\}. \end{aligned}$$
We assume that the time-step for the model is 1 year, and that the plant population is annual. Thus, we make the assumption that every plant behaves like the average plant. The seed bank population \(n(\cdot ,t)\) will evolve in the positive cone \(C[0,D]^{+}\), where \(D\) represents the deepest depth where a seed is both viable for germination and able to be disturbed, and the plant population \(p(t)\) evolves in the non-negative reals \(\mathbb {R}^{+}\). The dimensions of \(n(\cdot ,\cdot )\) are \(seeds(depth)^{-1}( area)^{-1}\), while the dimensions of \(p(\cdot )\) are \(plants(area)^{-1}\), with \(t = 0, 1, 2, \ldots \).

Between \(t\) and \(t+1\) we assume the following order of events: disturbance, redistribution of seeds, germination or dormancy, seed survival or plant recruitment, and production of new seeds. These events are assumed to occur immediately following the census of the entire seed bank population (both leftover ungerminated seeds and newly created seeds) from time \(t\). Thus, we consider only disturbances which occur after newly created seeds have been dispersed because disturbances prior to dispersal have a negligible effect on the seed bank (see, for example, Moody-Weis and Alexander’s 2007 study of wild sunflower). Furthermore, we model the disturbance as a single event in one time-step, which can be thought of as an average of the post dispersal disturbances to the population in a given year. Finally, we assume that seeds in the seed bank can only germinate in the event of a disturbance.

The entire seed bank population at time \(t\) and depth \(z\) is \(x(z)_{t} := n(z,t) + \kappa J(z)p(t)\), which is the sum of existing dormant seeds \(n(z,t)\) and the newly created seeds \(\kappa J(z)p(t)\), where \(\kappa \) seeds are created per plant, whose depth distribution is \(J\), which will be discussed in later paragraphs. We model disturbances at each time step \(t\) by the integration (in the depth variable \(y\)) of the entire seed bank population \(x(y)_{t}\) times a kernel function \(K(z,y,\omega _t)\), selected from a random sequence of kernels \(\{K(z,y,\omega _t)\}_{t=0}^{\infty }\), with each \(\omega _t \in \varOmega \). The dimension of \(K\) is \((depth)^{-1}\), which will become more apparent during its construction.

We will denote the sample space of all possible disturbances by \(\varOmega \). We let \(\omega \in \varOmega \) be determined by two disturbance characteristics: whether or not the population is disturbed and, if disturbed, how deep the disturbance affects the seed bank. We assume that there is a fixed positive integer \(T\) such that the seed bank is disturbed each time step with probability \(h\), unless there have been \(T\) time steps since the last disturbance, in which case the seed bank is disturbed with probability \(1\). We define \(\omega _{1}\) to be this (time-dependent) Bernoulli random variable, which is equal to unity in the event of a disturbance and zero otherwise. We define the depth of disturbance \(\omega _{2}\) as a random variable with probability density function \(\rho _2(\cdot )\) such that \(\text {supp}(\rho _2) = [\sigma ,D]\), where \(\sigma > 0\) is the minimum disturbance depth, with dimensions depth. Therefore, with the convention that not disturbing the seed bank population at all is the same thing as disturbing it to depth zero, the random variable \(\omega := \omega _{1} \omega _{2}\). The random variable \(\omega _{1}\) is dimensionless, while \(\omega _{2}\) has the dimensions of \(depth\), so \(\omega \) has the dimension of \(depth\).

We assume for biological purposes that \(T \gg 1\) is large and that \(\sigma \ll 1\) is small. Both of these assumptions are reasonable ecologically, since the probability of having \(T\) consecutive disturbance-free years is extremely small for large \(T\) and a small disturbance depth will affect so few seeds that it may as well be considered not a disturbance at all. Notice that the sequence \(\{w_t\}_{t=0}^{\infty }\) is \(iid\) when \(T = \infty \), which leads to our characterization of \(\{w_t\}_{t=0}^{\infty }\) as “roughly iid” in our setting.

In the event of a disturbance (\(\omega > 0\)), we assume that the seed bank population above the depth of disturbance \(\omega \) is uniformly distributed, while the remainder of the seed bank is roughly left alone (the precise definition is upcoming). A fraction of the entire seed bank then germinates and becomes seedlings (and later plants) according to a depth-dependent germination function. After germination has been determined, the remaining seed bank population then survives to the next time-step according to a depth-dependent survival function. We assume that germination and survival are probabilistically independent events.

In the event that there is no disturbance (\(\omega = 0\)), the seed bank population is left alone and no seeds are allowed to germinate. Since no seeds can germinate when there is no disturbance, the plant population is zero and the seed bank population is then multiplied by the same depth-dependent survival function.

We assume that, when \(\omega > 0\), the survival and germination functions, \(s(\cdot )\) and \(g(\cdot ,\omega )\), are and continuous functions on \([0,D]\) with maxima \(\overline{s}, \overline{g} < 1\) and minima \(\underline{s}, \underline{g} > 0\). When there is no disturbance we set \(g(\cdot ,0) = 0\), as germination cannot occur without a disturbance. When there is a disturbance (that is, \(\omega >0\)), we write \(g(z, \omega ) = g(z)\), since \(g\) is independent of \(\omega > 0.\) In our numerical examples, we will use functional forms from Mohler (1993), which assume that seed germination is a decreasing function of seed depth and seed survival is an increasing function of seed depth. We assume that \(s(\cdot )\) and \(g(\cdot ,\cdot )\), being probabilities, are dimensionless.

We assume that the plant recruitment step from seedling to plant is density dependent through the presence of contest competition. Let \(f\) be the function that takes the density of seedlings to the density of plants. This is the only component of the model that is density dependent (nonlinear). Contest competition in this setting models the idea that there are a fraction of competitors who obtain all of the resources they need to graduate from seedling to plant, while the remaining competitors all receive insufficient resources and die off (Anazawa 2012). We assume that the function \(f\) is continuous and bounded from \(\mathbb {R}^{+}\) into \(\mathbb {R}^{+}\), differentiable at \(0, f(0) = 0\), and if \(a>b>0\) then \(f(a) > f(b)\) and \(f(a)/a < f(b)/b\). These assumptions are consistent with the idea of monotonically diminishing per-capita returns and eventual saturation. Note that, because \(g(\cdot ,0) = 0\) and \(f(0) = 0\), when there is no disturbance there will be zero plants during that time-step, as sought. Also note that, while we make the distinction between seedlings and plants, we assume that the process from seed to seedling and the process from seedling to plant both occur in one time-step. Thus, we do not explicitly model seedlings in our model. The dimensions of \(f\) is \(plants\).

We will assume that seed production is density independent, i.e., each plant produces a constant \(\kappa \) seeds each time-step, with dimensions \(seeds(plant)^{-1}\). These seeds then enter the soil according to a depth-dependent function \(J(\cdot )\), which is a probability distribution that is positive and continuous throughout all of \([0,D]\) (with minimum \(\underline{J} > 0\) and maximum \(\overline{J}\)). Since \(J(\cdot )\) is a probability density function, it follows that \(\int _0^D J(z)\,\mathrm{d}z = 1\). We assume that \(J(\cdot )\) has the dimensions of \((depth)^{-1}\). We assume that these seeds enter the soil prior to any disturbance event. With this, the entire seed bank population \(x(z)_{t+1}\) at depth \(z \in [0, D]\) and time \(t+1\) solves
$$\begin{aligned} x(z)_{t+1}&= \mathop \int \limits _0^D s(z)(1-g(z,\omega _t))K(z,y,\omega _t) x(y)_t\, \mathrm{d}y \nonumber \\&+\, \kappa J(z) f\left( \mathop \int \limits _0^D \mathop \int \limits _0^D g(z,\omega _t)K(z,y,\omega _t)x(y)_t\, \mathrm{d}y\,\mathrm{d}z\right) , \end{aligned}$$
where \(t = 0,1,2, \ldots \) and \(x(\cdot )_0 \ne 0\) in \(C[0,D]^{+}\), the first term on the right-hand side of (1) is the dormant seed bank population \(n(z,t+1)\) at time \(t+1\) and the second term is the function \(\kappa J(z)\) multiplied by the plant population \(p(t+1)\) at time \(t+1\).

Up until this point we have been imprecise as to what form the kernel \(K(\cdot ,\cdot ,\omega )\) would take for each \(\omega \in \varOmega \). In the event that there is no disturbance at time \(t\) (\(\omega _t = 0\)) the kernel is the identity kernel \(K(z,y,0) = \delta (z-y)\) and the population projection operator is a multiplication operator, since the seed bank is left alone (prior to survival) in these time-steps. This multiplication operator in the infinite-dimensional space \(C[0,D]^{+}\) is not compact (Aubin 2000). In order to apply the results in Hardin et al. (1988), we need the population projection operator to be compact (we will see below that we really only need it to be compact at time steps \(t\) where \(w_t > 0\)).

To ensure that the disturbance kernels induce compact operators in the event of a disturbance, it is sufficient that \(K(\cdot , \cdot , \cdot )\) be in \(C^{1}([0,D]^{2} \times [\sigma , D])\), since if \(\omega > 0\) it is also true that \(\omega \ge \sigma \). We will also insist that there exist \( 0 < \underline{K}\) and \(\overline{K} < \infty \) such that
$$\begin{aligned} 0 < \underline{K} \le K(z,y, \omega ) \le \overline{K}, \end{aligned}$$
for all \(z,y \in [0,D]\) and \(\omega \in [\sigma , D]\).
We construct kernels \(K(\cdot ,\cdot ,\omega )\) that roughly distribute seeds uniformly above a level which depends on \(\omega \), while leaving seeds below this level roughly in place, see Eager et al. (2013a). We will construct \(K(\cdot ,\cdot ,\omega )\) by a limiting process using intermediate, piecewise-defined kernels \(\hat{K}(\cdot ,\cdot ,\omega , \epsilon _0)\), where \(\epsilon _0 >0\). We will start by insisting that \(\hat{K}(\cdot ,\cdot ,\omega , \epsilon _0)\) uniformly distributes seeds up until depth \(\omega \), that is,
$$\begin{aligned} \hat{K}(z,y,\omega , \epsilon _0) = 1/\omega \mathrm{~ for~} (z,y) \in [0,\omega ]^{2}. \end{aligned}$$
Next, we assume that \(\hat{K}(\cdot ,\cdot ,\omega , \epsilon _0)\) roughly leaves seeds alone for depths deeper than \(\omega \). The functional form we will use to model this is the probability density function for the normal distribution (as a function of \(z\)) with mean \(y\) and variance \(\epsilon _{0}^{2} > 0\), i.e., seeds are most likely to stay where they are, with very little deviation. For \(\epsilon _0\) sufficiently small, this normal distribution assigns very small values to \((z,y) \in ([0,\omega ]\times [\omega , D]) \cup ([\omega , D] \times [0, \omega ])\). Hence the kernel
$$\begin{aligned} \hat{K}(z,y,\omega , \epsilon _0) =\left\{ \begin{array}{l@{\quad }l} 1/\omega &{}: (z,y) \in [0,\omega ]^2 \\ (2\pi \epsilon _0^2)^{-1/2}\exp (-((z - y)/\epsilon _0)^2) &{}: (z,y) \in ([0,\omega ]^2)^{c}, \end{array} \right. \end{aligned}$$
where \(\omega \) is the depth of disturbance, roughly uniformly distributes the seeds in \([0,\omega ]\) and roughly leaves seeds in \([\omega , D]\) alone for sufficiently small \(\epsilon _0\), and will do so precisely as \(\epsilon _0 \rightarrow 0\). As such it follows that
$$\begin{aligned} \lim _{\epsilon _0 \rightarrow 0} || \mathop \int \limits _0^D\hat{K}(\cdot , y, \omega , \epsilon _0)x(y) \mathrm{d}y||_{\infty } \le ||x(\cdot )||_{\infty } \end{aligned}$$
for all \(x \in C[0,D]\) and \(\omega \in [\sigma , D]\). For the remainder of the paper we will suppress the dependence of \(\hat{K}\) and \(K\) on \(\epsilon _0\), as the remainder of the results only depend on \(\epsilon _0\) being strictly positive and sufficiently small.
It is easy to show that
$$\begin{aligned} \hat{K}(z,y,\omega ) \ge \underline{\hat{K}} = \min \{1/D, (2\pi \epsilon _0^2)^{-1/2}\exp (-(D/\epsilon _0)^2)\}>0 \end{aligned}$$
$$\begin{aligned} \hat{K}(z,y,\omega ) \le \overline{\hat{K}} = \max \{1/\sigma , (2\pi \epsilon _0^2)^{-1/2}\} \end{aligned}$$
for all \(\omega \in [\sigma , D]\) and \((z,y) \in [0,D]^2\). Furthermore, the kernel \(\hat{K}(\cdot ,\cdot ,\cdot )\) is piecewise-\(C^\infty \) on \([0,D]^2 \times [\sigma , D]\). Since \(C^1([0,D]^2 \times [\sigma , D])\) is dense in piecewise-\(C^\infty \) functions on \([0,D]^2 \times [\sigma , D]\), we can choose a new kernel \(K(\cdot , \cdot , \cdot )\) in \(C^1([0,D]^2 \times [\sigma , D])\) as close as we would like to \(\hat{K} (\cdot , \cdot , \cdot )\) on \([0,D]^2 \times [\sigma , D]\), and such that there exists \(\underline{K}\) and \(\overline{K}\) so that (2) is satisfied for all \((z,y) \in [0,D]^2\) and \(\omega \in [\sigma , D]\). It may not be possible to choose \(K\) to satisfy the inequality (3), but it is possible to choose \(\epsilon _0\) small enough and \(K\) close enough to \(\hat{K}\) such that
$$\begin{aligned} \overline{s}|| \mathop \int \limits _0^DK(\cdot , y, \omega )x(y) \mathrm{d}y||_{\infty } \le ||x(\cdot )||_{\infty }, \end{aligned}$$
where \(\overline{s}\) is the maximum of the survival function \(s(\cdot )\), which will be sufficient for our purposes. This is the kernel we use for the model (1).

It is straightforward to note that \(\hat{K}\) has the dimensions of \((depth)^{-1}\) and thus so does \(K\). With \(K\) defined as above, the sequence of population vectors \(\{x(\cdot )_t\}_{t = 0}^{\infty }\) from (1) evolves in \(C^{+}[0,D]\). We proceed by reformulating the model (1) into an abstract stetting in order to obtain the desired asymptotic results.

3 Abstract Formulation

3.1 Weak Convergence and Convergence in Distribution

For the theoretical results in this paper we will work with sequences of population vectors \(\{x_t\}_{t=0}^{\infty }\) in the Banach space \(C[0,D]\) of continuous real-valued functions over the interval \([0,D]\) with the sup norm. To define a partial ordering on \(C[0,D]\) we say that, for \(x_1,x_2 \in C[0,D]\), \(x_1 \ge x_2\) if \(x_1(z) \ge x_2(z)\) for all \(z \in [0,D]\). We will denote the zero function by \(0\) and the constant one function by \(1\!\!1\). Because biological populations are positive we will work with the positive cone \(C[0,D]^{+} := \{x \in C[0,D]: x \ge 0\}\) of \(C[0,D]\).

We now discuss what we mean when we say that the population sequence \(\{x_t\}_{t=0}^{\infty } \subset C[0,D]^{+}\) converges to a stationary distribution of populations as \(t \rightarrow \infty \). Let \(\mathcal {B}(C[0,D]^{+})\) be the family of Borel sets in \(C[0,D]^{+}\) and \(Z(C[0,D]^{+})\) be the family of Borel probability measures on \(C[0,D]^{+}\). A sequence of measures \(\{\mu _t\}_{t=0}^{\infty } \subset Z(C[0,D]^{+})\) is said to converge weakly to \(\mu ^{*} \in Z(C[0,D]^{+})\) if for all bounded, continuous, real-valued functionals \(\phi \) on \(C[0,D]^{+}\) we have that
$$\begin{aligned} \lim _{t\rightarrow \infty } \mathop \int \limits \phi \, \mathrm{d}\mu _t = \mathop \int \limits \phi \, \mathrm{d}\mu ^{*}. \end{aligned}$$
We will use the notation \(\mu _t \Rightarrow \mu ^{*}\) to denote weak convergence of the sequence \(\{\mu _t\}_{t=0}^{\infty }\) to \(\mu ^{*}\). A sequence of random populations \(\{x_t\}_{t=0}^{\infty } \subset C[0,D]^{+}\) is said to converge in distribution if the measures \(\{\mu _t\}_{t=0}^{\infty }\) associated with \(\{x_t\}_{t=0}^{\infty }\), defined by
$$\begin{aligned} \mu _t(A) = \Pr (x_t \in A), \; \; \; \; A \in \mathcal {B}(C[0,D]^{+}), \end{aligned}$$
converge weakly to some measure \(\mu ^{*} \in Z(C[0,D]^{+})\). For more on the convergence of probability measures over \(C[0,D]^{+}\) see Billingsley (1971), Billingsly (1995) and Hardin et al. (1988).

3.2 Reformulation of the Model (1)

In order to prove the desired weak convergence results for the sequences \(\{n(\cdot ,t)\}_{t=0}^{\infty }\) and \(\{p(t)\}_{t=0}^{\infty }\) modeling the seed bank and plant populations, respectively, we first need to prove the weak convergence of the sequence \(\{x_t\}_{t=0}^{\infty } \subset C^{+}[0,D]\), where
$$\begin{aligned} x(\cdot )_t := n(\cdot ,t) + \kappa J(\cdot )p(t). \end{aligned}$$
The population \(x_{t}\) solves the stochastic difference equation (with the dependence on space omitted)
$$\begin{aligned} x_{t+1} := H(\omega _{t},x_{t}) := A_0(\omega _t)x_t + bf(c(\omega _t)^{T}x_t), \end{aligned}$$
where \(\{\omega _t\}_{t=0}^{\infty } \subset \varOmega \). Here, \(b \in C^{+}[0,D]\) is the function \(\kappa J(\cdot ), A_0(\omega )\) are the linear operators from \(C^{+}[0,D]\) to itself defined by
$$\begin{aligned} A_0(\omega )u := \mathop \int \limits _0^D s(\cdot )(1-g(\cdot ,\omega ))K(\cdot ,y,\omega )u(y)\, \mathrm{d}y, \end{aligned}$$
and \(c^\mathrm{T}(\omega )\) are the functionals from \(C^{+}[0,D]\) into \(\mathbb {R}^{+}\) defined by
$$\begin{aligned} c(\omega )^\mathrm{T}u := \mathop \int \limits _0^D \mathop \int \limits _0^D g(z,\omega )K(z,y,\omega )u(y)\, \mathrm{d}y\, \mathrm{d}z, \end{aligned}$$
for \(u \in C^{+}[0,D]\). This formulation has been exploited for deterministic plant and plant-seed bank models in Rebarber et al. (2012), Townley et al. (2012) and Eager et al. (2013b).
If we can prove that the measures associated with the sequence \(\{x_t\}_{t=0}^{\infty }\) converge weakly to a measure \(\mu ^{*}\), independent of initial population, we can conclude that the measures associated with \(\{n(t)\}_{t=0}^{\infty }\), representing the seed bank population, and \(\{p(t)\}_{t=0}^{\infty }\), representing the plant population, converge weakly as well. To see this note that, from (1),
$$\begin{aligned} n(\cdot ,t+1)&= A_0(\omega _t, x_t)\end{aligned}$$
$$\begin{aligned} p(t+1)&= f(c^\mathrm{T}(\omega _t,x_t)). \end{aligned}$$
The maps in (5) and (6) are continuous maps from \(C[0,D]^{+} \rightarrow C[0,D]^{+}\) and \(C[0,D]^{+} \rightarrow \mathbb {R}^{+}\), respectively. Thus, if we define the measures
$$\begin{aligned} \hat{\mu }_1(A) := \mathop \int \limits _{\varOmega }\mu ^{*}(A_0^{-1}(\omega ,A))\; \mathrm{d}\omega , \; \; \; A \in \mathcal {B}(C[0,D]^{+}) \end{aligned}$$
$$\begin{aligned} \hat{\mu }_2(A) := \mathop \int \limits _{\varOmega }\mu ^{*}((c^\mathrm{T})^{-1}(\omega ,(f^{-1}(A))) \; \mathrm{d}\omega , \; \; \; A \in \mathcal {B}(\mathbb {R}^{+}), \end{aligned}$$
then \(\hat{\mu }_1\) and \(\hat{\mu }_2\) are the desired equilibrium probability measures associated with \(\{n(\cdot ,t)\}_{t=0}^{\infty }\) and \(\{p(t)\}_{t=0}^{\infty }\) (Folland 2006). These probability measures would thus capture the long-term behavior of the coupled populations individually.
The stochastic system (4) will satisfy the conditions of Theorems 4.2 and 5.3 in Hardin et al. (1988) (for suitable \(f\)) if the probability that \(\omega = 0\) is zero. However, in our system, this is not the case, since there are years where there is no disturbance. In particular, if \(\omega _t = 0\) for some \(t\), then for that \(t\)
$$\begin{aligned} x_{t+1} := H(0,x_t): = A_0(0)x_t , \end{aligned}$$
so there is no function \(\psi : [0,1) \rightarrow [0,1)\) such that \(\psi (v) > v\) for \(v \in (0,1)\) and
$$\begin{aligned} \psi (v) H(\omega ,x) \le H(\omega ,vx), \end{aligned}$$
for each \(\omega \in \varOmega \). That is because \(H(0,vx) =A_0(0)vx= vA_0(0)x = vH(0,x)\). Since the existence of such a \(\psi \) is a main ingredient used in the proofs in Hardin et al. (1988) we cannot apply the results in that paper directly to this system.
Given \(\{\omega _t\}_{t=0}^{\infty }\), let \(\mathcal{J} = \{t \in {\mathbb {N}} \mid \omega _{t} > 0\}\) be the subset of all time-steps for which there is a disturbance, which can be ordered to create a subsequence \(\{\hat{t}\}_{t=0}^{\infty } \in \mathbb {N}\cup \{0\}\). We create this subsequence in the following way: If \(\omega _0 > 0\) (there is a disturbance at \(t=0\)) define \(\hat{t} \in \mathcal {J}\) to be the timestep of the \((t+1)\mathrm{st}\) disturbance. If \(\omega _0 = 0\) (there is no disturbance at \(t=0\)), let \(\hat{0} = 0\) and define \(\hat{t} \in \mathcal {J}\) to be the timestep of the tth disturbance. To allow us to use the results in Hardin et al. (1988) we study the population on the sequence of time-steps \(\{\hat{t}\}_{\hat{t} \in \mathcal{J}}\). We note that, when \(\omega _t = 0\), in addition to being linear, the model (1) is also completely deterministic. To see this, note that, when \(\omega _t = 0,\) the model (1) becomes
$$\begin{aligned} x(z)_{t+1}&= \mathop \int \limits _0^D s(z)(1-g(z,0))K(z,y,0) x(y)_t\, \mathrm{d}y \nonumber \\&+\, \kappa J(z) f\left( \mathop \int \limits _0^D \mathop \int \limits _0^D g(z,0)K(z,y,0)x(y)_t\, \mathrm{d}y\, \mathrm{d}z\right) ,\nonumber \\&= s(z)x(z)_t\,dy + \kappa J(z) f(0)\nonumber \\&= s(z)x_t(z) \end{aligned}$$
by the definition of \(K(\cdot ,\cdot ,0)\) and the fact that \(g(z,0) = 0\) for all \(z \in [0,D]\). Therefore, if we let \(\tau _t\) be the number of time-steps between (disturbance) time-steps \(\hat{t}\) and \(\widehat{t+1}\) and only track \(x_{{t}}\) for those time steps \(\hat{t}\) where there are disturbances (i.e. gather together all non-disturbance time-steps into one \(\hat{t}\) time-step), then the model (4) becomes
$$\begin{aligned} x\,_{\widehat{t+1}} := \hat{H}(\omega _{\hat{t}},x_{\hat{t}}): = \hat{A}_0(\omega _{\hat{t}})x_{\hat{t}} + bf(\hat{c}(\omega _{\hat{t}})^\mathrm{T}x_{\hat{t}}), \end{aligned}$$
$$\begin{aligned} \hat{A}_0(\omega )u := \mathop \int \limits _0^D s(\cdot )(1-g(\cdot ))K(\cdot ,y,\omega _2)s(y)^{\tau }u(y)\, \mathrm{d}y, \end{aligned}$$
$$\begin{aligned} \hat{c}(\omega )^\mathrm{T}u := \mathop \int \limits _0^D \mathop \int \limits _0^D g(z)K(z,y,\omega _2)s(y)^{\tau }u(y)\,\mathrm{d}y\, \mathrm{d}z. \end{aligned}$$
With this reformulation our stochastic process is now the sequence \(\{(\omega _2)_{\hat{t}}, \tau _{\hat{t}}\}_{\hat{t} \in \mathcal{J}}\), where \((\omega _2)_{\hat{t}}\) is the depth of disturbance at the time-step \(\hat{t}\) and the random variable \(\tau _{\hat{t}}\) is the number of time-steps between the disturbance at time \(\hat{t}\) and \(\widehat{t+1}\) (with dimensions of \((time)\)). We know through the original construction of the model that \((\omega _2)_{\hat{t}}\) has the probability density function \(\rho _2(\cdot )\) for each \(\hat{t}\). It is straightforward to see that \(\tau \) has the geometric distribution, truncated at \(T\), i.e.
$$\begin{aligned} \Pr (\tau = \tau _0) := \left\{ \begin{array}{l@{\quad }l} h(1-h)^{\tau _0} &{}: \tau _0 = 0, 1, \ldots ,T - 1\\ (1 - h)^\mathrm{T} &{}: \tau _0 = T. \end{array} \right. \end{aligned}$$
It also follows that \((\omega _2)_{\hat{t}}\) and \(\tau _{\hat{t}}\) are probabilistically independent for each \(\hat{t}\). With this we will define the stochastic process \(\{\hat{\omega }_{\hat{t}}\}_{\hat{t} \in \mathcal{J}} = \{(\omega _2)_{\hat{t}}, \tau _{\hat{t}}\}_{\hat{t}\in \mathcal{{J}}}\), where \(\{\hat{\omega }_{\hat{t}}\}_{\hat{t}\in \mathcal{{J}}}\) is now a sequence of \(iid\) random variables coming from the space \(\hat{\varOmega }\) of all possible environmental states defining the stochastic process. Note that the probability spaces \(\hat{\varOmega }\) and \(\varOmega \) contain the same relevant probabilistic information, as the values of \(\omega _2\) when \(\omega = 0\) do not influence the population. Also notice that, since \(\omega _{\hat{t}} \ne 0\) for all \(\hat{t}\), it follows that \(g(z, \omega _{\hat{t}})\) is simply \(g(z)\) for all \(\hat{t}\).
If we can prove that the probability measures associated with the sequence \(\{x_{\hat{t}}\}_{\hat{t}\in \mathcal {J}}\) converge weakly to the measure \(\mu ^{*}\), independent of initial population, we will have that the probability measures associated with the sequence \(\{x_t\}_{t=0}^{\infty }\) (that explicitly includes non-disturbance time-steps) converge weakly as well. To see this, note that the maps \(\gamma _{\tau }:C[0,D]^{+} \rightarrow C[0,D]^{+}, \tau = 1,2, \ldots ,T\) defined by
$$\begin{aligned} \gamma _{\tau }(u)(\cdot ) := s(\cdot )^{\tau }u(\cdot ) \end{aligned}$$
are continuous. With this we can define the measures
$$\begin{aligned} \hat{\mu }_{1,\tau }(A) := \hat{\mu }_1((\gamma ^{\tau })^{-1}(A))), \; \; \; A \in \mathcal {B}(C[0,D]^{+}), \end{aligned}$$
where \(\hat{\mu }_1\) (as a function of \(\mu ^{*}\)) is from (7). It follows that the equilibrium measure associated with the seed bank population \(n(\cdot ,t)\) is, due to the Law of Total Probability (Billingsly 1995),
$$\begin{aligned} \mu ^{*}_1(A) := p_0\hat{\mu }_1(A) + p_1\hat{\mu }_{1,1}(A) + \cdots + p_{T-1}\hat{\mu }_{1,T - 1}(A) + p_T\hat{\mu }_{1,T}(A), \end{aligned}$$
for \(A \in \mathcal {B}(C[0,D]^{+})\), where the vector of probabilities \([p_0, \; p_1, \ldots , p_T]^\mathrm{T}\) represents the stationary distribution of the “age” of the seed bank, or the number of time-steps since the last disturbance, which can be computed by finding the leading right eigenvector of the \(\mathrm{T} \times \mathrm{T}\) transition matrix
$$\begin{aligned} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} h &{} h &{} h &{} \ldots &{} h &{} 1 \\ 1-h &{} 0 &{} 0&{} \ldots &{} 0 &{} 0 \\ 0 &{} 1 - h &{} 0 &{} \ldots &{}0 &{} 0 \\ \vdots &{} \vdots &{} \ddots &{} \ldots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 1-h &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \ldots &{} 1-h &{} 0 \end{array} \right) . \end{aligned}$$
It is not difficult to show that
$$\begin{aligned} p_T = \frac{h}{(1-h)((1-h)^{-(T + 1)} - 1)}, \end{aligned}$$
and that \(p_i = (1 - h)^{T-i}p_T\) for \(i = 0, 1, 2, \ldots , T - 1\).
Also recall that the probability of disturbance is \(h\), and thus we can alter the measure \(\hat{\mu }_2\) from (8) associated with the plant population by defining
$$\begin{aligned} \mu ^{*}_2(A) := h\hat{\mu }_2(A) + (1 - h)\delta _{\{0\}}(A), \end{aligned}$$
for \(A \in \mathcal {B}(\mathbb {R}^{+})\), where \(\delta _{\{0\}}\) is the Dirac mass centered at the zero function, to obtain the asymptotic measure associated with the plant population \(p(t)\) in model (4). Here, we use the fact that the plant population will be zero with at least the probability of no disturbance \(1-h\).

With this discussion in mind, we will establish the long-term behavior of the sequence \(\{x_{\hat{t}}\}_{t=0}^{\infty }\), which will imply the long-term behavior of the sequences \(\{n(\cdot ,t)\}_{t=0}^{\infty }\) and \(\{p(t)\}_{t=0}^{\infty }\).

3.3 Mathematical Conditions for the Model (9)

For notational simplicity, the remainder of this paper we will denote \(\{\hat{\omega }_t\}_{t=0}^{\infty }, \hat{\varOmega }\), \(\hat{H}(\cdot , \cdot ), \hat{A}_0(\cdot )\), and \(\hat{c}^\mathrm{T}(\cdot )\) by \(\{\omega _t\}_{t=0}^{\infty }, \varOmega , H(\cdot , \cdot )\), \(A_0(\cdot )\) and \(c^\mathrm{T}(\cdot )\), respectively.

Let the initial state \(x_0 \in C[0,D]^{+}\setminus {\{0\}}\) be a random variable. Since \(\{\omega _t\}_{t=0}^{\infty }\) is a sequence of iid random variables, model (9) defines \(\{x_t\}_{t=0}^{\infty }\) as a Markov sequence of random variables.

We will now summarize the assumptions for our model (9). These assumptions are very similar to those in Section 3.1 of Hardin et al. (1988).
  1. (A1)

    \(\{\omega _t\}_{t = 0}^{\infty } = \{(\omega _2)_t, \tau _t\}_{t=0}^{\infty }\) is a sequence of \(iid\) random variables from the probability space \(\varOmega \), with \((\omega _2)_t \in [\sigma , D]\) and \(\tau _t \in \{0, 1, 2, \ldots , T\}\) for all \(t\).

  2. (A2)
    The functions \(s, g \in C[0,D]\), and have positive lower bounds \(\underline{s}, \underline{g}\) and upper bounds \(\overline{s}, \overline{g}\), respectively. The function \(K(\cdot , \cdot , \omega _2)\) has positive lower bound \(\underline{K}\) and upper bound \(\overline{K}\), and
    $$\begin{aligned} \overline{s}||\mathop \int \limits _0^D K(\cdot , y, \omega _2)x(y) \mathrm{d}y||_{\infty } \le ||x(\cdot )||_{\infty } \end{aligned}$$
    for all \(x \in C[0,D]^{+}\) and \(\omega _2 \in [\sigma , D]\). As a result, the kernel functions \(s(\cdot )(1-g(\cdot ))K(\cdot , \cdot , \omega _2)s(\cdot )^{\tau }\) and \(g(\cdot ))K(\cdot , \cdot , \omega _2)s(\cdot )^{\tau }\) in the definitions of \(A_0(\omega )\) and \(c(\omega )\) from (9), respectively, are continuous and uniformly bounded above and below (away from zero), and these bounds are independent of \(\omega \).
  3. (A3)

    The function \(K \in C^1([0,D]^2 \times [\sigma , D])\)

  4. (A4)

    The vector \(b\) in (9) is given by the function \(\kappa J \in C[0,D]^{+}\), where \(0 < \underline{J} \le J(z) \le \overline{J}\) for all \(z \in [0,D]\).

  5. (A5)

    The function \(f\) is continuous and uniformly bounded from \(\mathbb {R}^{+}\) into \(\mathbb {R}^{+}\), differentiable at \(0\), \(f(0) = 0\), and if \(a>b>0\) then \(f(a) > f(b)\) and \(f(a)/a < f(b)/b\).

The assumptions (A3), (A4) and (A5) and (12), as well as the continuity and boundedness properties of \(s(\cdot ), g(\cdot )\), and \(K(\cdot , \cdot , \cdot )\), have been explicit assumptions since the introduction of the model in Sect. 1, and assumption (A1) was made true by the reformulation of the model to (9). To see that the additional information in (A2) is true note that, for each \(\omega \in \varOmega \), the kernel function used in defining the operator \(A_0(\omega )\) is \(s(x)(1-g(x))K(x,y,\omega _2)s(y)^{\tau }\), which is continuous, as it is the product of continuous functions. Furthermore we have that, for all \((y,z) \in [0,D]^2\) and \(\omega \in \varOmega \),
$$\begin{aligned} 0 < \underline{s}^{T + 1}(1 - \overline{g})\underline{K} \le s(z)(1-g(z))K(z,y,\omega _2)s(z)^{\tau } \le \overline{s}(1 -\underline{g})\overline{K}, \end{aligned}$$
thus the kernel defining the operator \(A_0(\omega )\) is bounded and uniformly bounded away from \(0\) for each \(\omega \in \varOmega \), with upper and lower bounds independent of \(\omega \in \varOmega \).
Similarly, the kernel function used in defining the functional \(c(\omega )\) is \(g(z)K(z,y,\omega _2)\)\(s(z)^{\tau }\), which is continuous, and it follows that
$$\begin{aligned} 0 < \underline{s}^\mathrm{T}\underline{g}\underline{K} \le g(z)K(z,y,\omega _2)s(z)^{\tau } \le \overline{g}\overline{K}, \end{aligned}$$
so the kernel defining the functional \(c(\omega )\) is bounded and strictly positive for all \(\omega \in \varOmega \), with the upper and lower bounds independent of \(\omega \).

Therefore, the model (9) for the plant-seed bank dynamics of a disturbance specialist satisfies assumptions (A1)–(A5).

3.4 Properties of \(H\)

Suppose that the assumptions (A1), (A2), (A3), (A4), and (A5) are satisfied for \(H(\cdot ,\cdot )\) given by the model (9).

The following properties of \(H(\cdot , \cdot )\) will be needed for the proof of Theorem 1. These properties are similar to those in Section 3.4 of Hardin et al. (1988) and will be proved for our model as a part of the proof of Theorem 1 in Sect. 4.1.
  1. (H1)

    For each \(\omega \in \varOmega \), \(H(\omega ,\cdot )\) is a continuous, monotone map from \(C[0,D]^{+}\) to itself such that \(H(\omega ,x) = 0\) if and only if \(x = 0 \in C[0,D]^{+}\).

  2. (H2)
    There exists some \(M > 0\) such that for all \(x \in C[0,D]^{+}\) and \(\omega \in \varOmega \),
    1. (a)

      \(||H(\omega , x)||_{\infty } \le M\) whenever \(||x||_{\infty } \le M.\)

    2. (b)
      There is some time-step \(t\) (which depends on \(x_0\)) \(\in \mathbb {N}\) such that
      $$\begin{aligned} ||H(\omega _t, x_t)\circ H(\omega _{t-1}, x_{t-1})\circ \cdots \circ H(\omega _0, x_0)||_{\infty } < M. \end{aligned}$$
    3. (c)
      There is some \(m > 0\) such that
      $$\begin{aligned} H(\omega ,M1\!\!1) \ge m1\!\!1, \end{aligned}$$
      for all \(\omega \in \varOmega \).
  3. (H3)
    For \(M\) as in (H2) let
    $$\begin{aligned} B_M := \{x \in C[0,D]^{+}: ||x||_{\infty } \le M \}. \end{aligned}$$
    There is some compact set \(\mathcal {D} \subset C[0,D]^{+}\) such that \(H(\omega ,B_M) \subset \mathcal {D}\) for all \(\omega \in \varOmega \). There also exists an \(\eta > 0\) such that
    $$\begin{aligned} H(\omega ,B_M) \subset \{x \in C[0,D]^{+}: x \ge \eta ||x||_{\infty }1\!\!1\} := m_{\eta } \end{aligned}$$
    for all \(\omega \in \varOmega .\)
  4. (H4)
    There exists some \(h >0\) such that
    $$\begin{aligned} ||H(\omega ,x)||_{\infty } \le h ||x||_{\infty }, \end{aligned}$$
    for all \(\omega \in \varOmega \) and \(x \in C[0,D]^{+}\).
  5. (H5)
    For each \(M_0 > 0\) there exists a function \(\psi :(0,1] \rightarrow (0,1]\) such that \(\psi (v) > v\) for \(v \in (0,1)\) and
    $$\begin{aligned} \psi (v)H(\omega ,x) \le H(\omega ,vx), \end{aligned}$$
    for all \(\omega \in \varOmega \) and \(x \in C[0,D]^{+}\) for which \(x \in [M_0,M]\), where \(M\) is from (H2).
  6. (H6)

    The operator \(H(\omega ,\cdot )\) is Fréchet differentiable at \(0 \in C[0,D]^{+}\) for each \(\omega \in \varOmega \). Let \(A(\omega )\) be the Fréchet derivative of \(H(\omega , \cdot )\) at \(0 \in C[0,D]^{+}\).

  7. (H7)
    There exists a function \(\nu :\mathbb {R}^{+} \rightarrow [0,1]\) such that \(\lim _{u\rightarrow 0^{+}}\nu (u)=1\) and such that
    $$\begin{aligned} \nu (D\overline{K}||x||_{\infty })A(\omega )x \le H(\omega ,x) \le A(\omega )x, \end{aligned}$$
    for \(x \in C[0,D]^{+}\) and \(\omega \in \varOmega \).

4 Results

4.1 Main Asymptotic Result

We will assume that the initial population \(x_0 \ne 0\) with probability one. We will show that the measures \(\{\mu _t\}_{t=0}^{\infty }\) associated with the population \(\{x_t\}_{t=0}^{\infty }\) converge weakly to some measure \(\mu ^{*} \in Z(C[0,D]^{+})\) which is independent of \(x_0\). The measure will either be concentrated at \(0 \in C[0,D]^{+}\) (extinction) or will be supported entirely in \(C[0,D]^{+}\setminus \{0\}\) (persistence). This dichotomy of results will depend entirely on the family \(\{A(\omega )\}_{\omega \in \varOmega }\) of Fréchet derivatives of the family of operators \(\{H(\omega ,\cdot )\}_{\omega \in \varOmega }\).

$$\begin{aligned} \lambda := \lim _{t\rightarrow \infty }||A(\omega _t) \circ A(\omega _{t-1}) \circ \cdots \circ A(\omega _0)1\!\!1||_{\infty }^{1/t}, \end{aligned}$$
which is independent of the path \(\{\omega _j\}_{j=0}^{t}\), by the Kingman Subadditive Ergodic Theorem (Kingman 1973). The following theorem says that, as \(t \rightarrow \infty \), the population goes extinct with probability one if \(\lambda < 1\) or persists with probability one if \(\lambda > 1\). \(\lambda \) represents the growth rate of the population when the population is rare, therefore, for the remainder of the paper, we will refer to \(\lambda \) as the rare growth rate.

Theorem 1

Suppose (A1), (A2), (A3), (A4), and (A5) for the model (9) are satisfied and \(x_0 \ne 0 \in C[0,D]^{+}\) with probability \(1\). Then \(x_t\) converges in distribution to a stationary distribution \(\mu ^{*}\), independent of \(x_0\), such that either \(\mu ^{*}(\{0\}) = 0\) or \(\mu ^{*}(\{0\}) = 1\). If \(\lambda > 1\), then \(\mu ^{*}(\{0\}) = 0\) and if \(\lambda < 1\), then then \(\mu ^{*}(\{0\}) = 1\).


To prove the result we simply need to verify (H1)–(H7), as the proof will then follow directly from that in Hardin et al. (1988). The continuity and monotonicity of \(H(\omega , \cdot ): C[0,D]^{+} \rightarrow C[0,D]^{+}\) follow from the continuity and strict positivity of \((A_0(\omega ), b, c^\mathrm{T}(\omega ))\) and the continuity and non-negativity of \(f\). The condition that \(H(\omega ,x) = 0\) if and only if \(x = 0\) follows from the linearity and strict positivity of \(A_0(\omega )\) and \(c^\mathrm{T}(\omega )\) and the fact that \(f(0) = 0\). Thus (H1) is verified.

For (H2) (a) and (b) note that, using (A2) (specifically the bound (12)) and (A5)
$$\begin{aligned} ||H(\omega ,x)||_{\infty } \le (1-\underline{g})||x||_{\infty } + \kappa \overline{J}\alpha , \end{aligned}$$
where \(\alpha := \sup \{f(z): z \in \mathbb {R}^{+}\}\), which is finite by assumption. Iterating the model (9) forward in time it follows that, for all \(t \in \mathbb {N}\)
$$\begin{aligned} ||x_{t+1}||_{\infty } \le \theta ^t||x_t||_{\infty } + \frac{\kappa \overline{J}\alpha }{1 - \theta }, \end{aligned}$$
where \(\theta := (1- \underline{g}) < 1\). Choose \(M > {\kappa \overline{J} \alpha }/(1- \theta )\) and (H2) (a) and (b) follow.
To prove (H2) (c) notice that the bound (14) from (A2) implies that
$$\begin{aligned} H(\omega ,M1\!\!1)&\ge \kappa \underline{J} f\left( M\mathop \int \limits _0^D \mathop \int \limits _0^D g(z)K(z,y,\omega _2)s(y)^{\tau }\, \mathrm{d}y\, \mathrm{d}z\right) 1\!\!1\nonumber \\&\ge \kappa \underline{J} f(M\underline{g}\underline{K}\underline{s}^\mathrm{T}D^2)1\!\!1 := m1\!\!1.\nonumber \end{aligned}$$
To prove (H3), note that it follows from \((H2)\) that if \(x \in B_M\) we have that \(H(\omega , x) \in B_M\) for all \(\omega \in \varOmega \). In other words, \(H(\omega , B_M) \subseteq B_M\). By (A3) we know that \(K \in C^1([0,D]^2 \times [\sigma , D])\). Therefore, there exists a constant \(M_e\) so that
$$\begin{aligned} \left| \frac{\partial K(z,y,\omega _2)}{\partial z}\right| \le M_e \mathrm{~for~} (z,y,\omega _2) \in [0,D]^2 \times [\sigma , D]. \end{aligned}$$
Thus, for \(x \in B_M\) and
$$\begin{aligned} F(z,\omega ) := \mathop \int \limits _0^D K(z, y, \omega _2)s(y)^{\tau }x(y)\, \mathrm{d}y \end{aligned}$$
we have, using (16), that
$$\begin{aligned} \left| \frac{\partial F(z,\omega )}{\partial z}\right| \le D M_e M \mathrm{~for~} (z,y) \in [0,D]^2 \mathrm {~and~} \omega \in \Omega . \end{aligned}$$
In particular, (17) proves that for \(z_1, z_2 \in [0,D]\) and \(\omega \in \varOmega \),
$$\begin{aligned} |F(z_1, \omega ) - F(z_2, \omega )| \le DM_e M |z_1 - z_2|, \end{aligned}$$
which is independent of \(\omega \). Hence the set
$$\begin{aligned} \left\{ \mathop \int \limits _0^D K(\cdot , y, \omega _2) s(y)^{\tau }x(y)\, \mathrm{d}y \mid \omega _2 \in [\sigma , D], x \in B_M\right\} \end{aligned}$$
is equicontinuous in \(C[0,D]\). The equicontinuity of \(\{H(\omega , x)\}_{\omega \in \varOmega }\) follows then from the fact that \(s(\cdot ), g(\cdot ), J(\cdot )\), and \(f(\cdot )\) are continuous, bounded and, most importantly, independent of \(\omega = (\omega _2, \tau )\). Thus, since \(\{H(\omega , x)\}_{\omega \in \varOmega }\) is a uniformly bounded and equicontinuous family of functions, the Arzela–Ascoli theorem guarantees the existence of a compact set \(\mathcal {D} \subseteq B_M\) such that \(H(B_M) \subseteq \mathcal {D}\).
Note that (A5) implies that \(f(z) \ge {f(z_0)z}/{z_0}\) for all \(z \le z_0\) and \(f(z) \le f'(0)z\) for all \(z \ge 0\). Now observe that, for the seed production \(z_0\) elicited by the maximum possible population \(M1\!\!1\),
$$\begin{aligned} z_0(\omega ) = \mathop \int \limits _0^{D} g(z)\mathop \int \limits _0^{D} K(z,y,\omega _2)s(y)^\tau M1\!\!1(y)\, \mathrm{d}y \, \mathrm{d}z > D^2\underline{s}^\mathrm{T}\underline{g}\underline{K}M, \end{aligned}$$
we have, using the inequality \(f(z) \ge {f(z_0)z}/{z_0}\) for all \(z \le z_0\) and the bound (14) from (A2),
$$\begin{aligned} H(\omega , x) \ge b f(c^\mathrm{T}(\omega )x) \ge b f(z_0)\frac{c^\mathrm{T}(\omega )x}{z_0} \ge \kappa \underline{J}\underline{s}^\mathrm{T}\underline{g}\underline{K}D\frac{f(z_0)}{z_0}\mathop \int \limits _0^Dx(z)\, \mathrm{d}z1\!\!1, \end{aligned}$$
for all \(\omega \in \varOmega \). Similarly, using the inequality \(f(z) \le f'(0)z\) for all \(z \ge 0\) and the bounds (13) and (14) from (A2),
$$\begin{aligned} H(\omega , x) \le (\overline{s}(1-\underline{g})\overline{K} + \overline{g}\overline{K}\kappa \overline{J}f'(0))\mathop \int \limits _0^Dx(z)\, \mathrm{d}z1\!\!1, \end{aligned}$$
for all \(\omega \in \varOmega \), which implies that
$$\begin{aligned} ||H(\omega , x)||_{\infty } \le (\overline{s}(1-\underline{g})\overline{K} + \overline{g}\overline{K}\kappa \overline{J}f'(0))\mathop \int \limits _0^Dx(z)\, \mathrm{d}z. \end{aligned}$$
Combining (20) with (19) we have that
$$\begin{aligned} H(\omega ,x) \ge \frac{\underline{s}^\mathrm{T}f(z_0)\underline{g}\underline{K}\kappa \underline{JD}||H(\omega ,x)||_{\infty }1\!\!1}{z_0\overline{s}(1-\underline{g})\overline{K} + \overline{g}\overline{K}\kappa \overline{J}f'(0)} := \eta ||H(\omega ,x)||_{\infty }1\!\!1, \end{aligned}$$
and (H3) is established.

For (H4) let \(h := (1- \underline{g}) + \kappa \overline{J}\overline{g}(\overline{s})^{-1}Df'(0)\). Since (A5) implies again that \(f(z) \le f'(0)z\) the property follows.

For (H5) note that for \(v \in (0,1)\), \(vf(z) < f(vz)\), thus \(\frac{f(vz)}{f(z)} > v\) for \(v > 0\). Let \(t(v) := \min \{\frac{f(vz)}{f(z)}\}\), which is continuous with \(t(v) > v\) for \(v \in (0,1)\). Define the function
$$\begin{aligned} \psi (v) : = \frac{t(v)\underline{\Theta } + v }{\underline{\Theta } + 1}, \end{aligned}$$
$$\begin{aligned} \underline{M} := \inf _{\omega \in \varOmega }\{c^\mathrm{T}(\omega )1\!\!1M_0\} \end{aligned}$$
$$\begin{aligned} \underline{\varTheta } : = \frac{\kappa \underline{J} f(\underline{M})}{\overline{s}(1 - \underline{g})\overline{K}}. \end{aligned}$$
It is easy to show that \(\psi (v) > v\), since \(t(v) > v\). Let
$$\begin{aligned} \varTheta :=\frac{\kappa J(z) f(c^\mathrm{T}(\omega )x)}{s(z)(1 - g(z))K(z,y, \omega _2)s(y)^{\tau }}. \end{aligned}$$
The function
$$\begin{aligned} \phi (\varTheta , v) := \frac{t(v)\varTheta + v}{\varTheta + 1} \end{aligned}$$
is increasing in \(\varTheta \), and \(\varTheta \ge \underline{\varTheta }\) from the bounds (13) and (14)). Thus, it follows that
$$\begin{aligned} \psi (v) = \phi (\underline{\varTheta }) \le \phi (\varTheta , v)=\frac{t(v)\kappa J(z)f(c^\mathrm{T}(\omega )x) + v\; s(z)(1 - g(z))K(z,y, \omega _2)s(y)^{\tau }}{\kappa J(z)f(c^\mathrm{T}(\omega )x) + s(z)(1 - g(z))K(z,y, \omega _2)s(y)^{\tau }} \end{aligned}$$
for all \((z,y) \in [0,D]^2, \omega \in \varOmega \) and \(x \in [M_0,M]\). Therefore,
$$\begin{aligned} H(\omega ,vx)&= \mathop \int \limits _0^Ds(z)(1 - g(z))K(z,y, \omega _2) s(y)^{\tau }v\; x(y)\, \mathrm{d}y \\&+\, \kappa J(z)f(c^\mathrm{T}(\omega ) v x)\\&\ge v \mathop \int \limits _0^Ds(z)(1 - g(z))K(z, y, \omega _2) s(y)^{\tau }x(y)\, \mathrm{d}y \\&+\, t(v)\kappa J(z)f(c^\mathrm{T}(\omega ) x)\\&\ge \psi (v)\left( \mathop \int \limits _0^Ds(z)(1 - g(z))K(z, y, \omega _2) s(y)^{\tau }x(y)\, \mathrm{d}y + \kappa J(z)f(c^\mathrm{T}(\omega )x)\right) \\&= \psi (v) H(\omega , x). \nonumber \end{aligned}$$
Thus \(H(\omega , vx) \ge \psi (v) H(\omega , x)\) for all \(x \in C[0,D]^{+}\) and \(\omega \in \varOmega \).
The Fréchet differentiability at \(x = 0 \in C[0,D]^{+}\) follows from differentiability of \(f(z)\) at \(z =0\) and the fact that \(c^\mathrm{T}(\omega ) = 0\) for all \(\omega \in \varOmega \), establishing (H6). The Fréchet derivative of \(H(\omega ,\cdot )\) is thus
$$\begin{aligned} A(\omega ) = (A_0(\omega ) + f'(0)bc^\mathrm{T}(\omega )). \end{aligned}$$
For (H7) define
$$\begin{aligned} \nu (z) = \left\{ \begin{array}{l@{\quad }l} 1 &{} z =0\\ \frac{f(z)}{zf'(0)} &{} z >0.\end{array}\right. \end{aligned}$$
It follows from assumption (4) that \(\nu \) is decreasing and \(\lim _{z\rightarrow 0^{+}}\nu (z) = 1\). Furthermore, for \(x \ne 0\),
$$\begin{aligned} f(c(\omega )^\mathrm{T}x)&= \nu (c^\mathrm{T}(\omega )x)c^\mathrm{T}(\omega )xf'(0)\\&\ge \nu \left( D\overline{g}\overline{K}||x||_{\infty }\right) c^\mathrm{T}(\omega )xf'(0). \end{aligned}$$
Thus, since \(f(z) \le f'(0)z\) we have
$$\begin{aligned} \nu \left( D\overline{K}||x||_{\infty }\right) A(\omega )x \le H(\omega ,x) \le A(\omega )x. \end{aligned}$$
The remainder of the proof follows from Hardin et al. (1988) \(\square \)

4.2 Establishing the Asymptotic Properties of the Original, Individual Populations \(\{p(t)\}_{t=0}^{\infty }\) and \(\{n(\cdot ,t)\}_{t=0}^{\infty }\)

Theorem 1 establishes the weak convergence of the measures \(\{\mu _t\}_{t=0}^{\infty }\) associated with the sequence \(\{x_t\}_{t=0}^{\infty } =\{n(\cdot ,t) + cJ(\cdot )p(t)\}_{t=0}^{\infty }\) when we assume that the \(t\)’s only represent disturbance years. The following corollary, which follows from the arguments in Sect. 3.2, recovers the asymptotic properties of the individual sequences \(\{n(\cdot ,t)\}_{t=0}^{\infty }\) and \(\{p(t)\}_{t=0}^{\infty }\).

Corollary 1

Suppose (A1), (A2), (A3), (A4) and (A5) for the model (9) are satisfied and \(x_0 = n(\cdot ,0) + cJ(\cdot )p(0) \ne 0 \in C[0,D]^{+}\) with probability \(1\). Then \(n(\cdot ,t)\) and \(p(t)\) converge in distribution to the stationary distributions \(\mu ^{*}_1\) and \(\mu ^{*}_2\) from (10) and (11), respectively, independent of \(n(\cdot ,0)\) and \(p(0)\).

5 Example

To illustrate how the results in this paper can be applied to a model for a disturbance specialist plant and its seed bank we will need to specify a probability distribution \(\rho _{2}\) for the disturbance depth \(\omega _{2}\), as well as functional forms for \(s, g, J\), and \(f\). This example is motivated by the studies in Alexander and Schrag (2003), Alexander et al. (2009) and Moody-Weis and Alexander (2007) for wild sunflower (Helianthus annuus).

We will assume that the depth of disturbance \(\omega _{2}\) is distributed via a (truncated) exponential distribution with mean \(\rho \). The probability density function is truncated in such a way so that all time-steps for which the depth of disturbance \(\omega _{2}\) would have been less than \(\sigma \) would be assigned the value \(\sigma \) and that all time-steps for which \(\omega _{2}\) would have been greater than \(D\) would be assigned the value \(D\). We will call \(\rho \) the mean depth of disturbance (although the true mean may not actually equal \(\rho \)).

We will follow the assumptions used in Eager et al. (2013a) and use functional forms for \(s\) and \(g\) introduced in Mohler (1993). For the survival probability of the seeds in the seed bank we make the simplifying assumptions that this fraction only depends on the seed’s depth in the seed bank, that seeds survive at their lowest rates near the surface of the soil (due largely to seed predation) and that the likelihood of survival increases as seed depth increases. The survival function is given by
$$\begin{aligned} s(z) := s_0(1 - \exp (-b_0z)), \end{aligned}$$
where \(s_{0} \in (0, 1)\) is the maximum possible survival probability of a seed (which is dimensionless) and \(b_{0}\) models the incremental gain in survival probability that occurs through an incremental increase in seed depth (with dimensions \((depth)^{-1}\)).
We assume that germination only occurs in the presence of a disturbance, i.e., the germination probability is a function of \(\omega \), i.e.
$$\begin{aligned} g(z,\omega ) := \left\{ \begin{array}{lr} g_p(z) &{} : \omega \ne 0\\ 0 &{} : \omega = 0. \end{array} \right. \end{aligned}$$
We assume that \(g_{p}\) has the functional form
$$\begin{aligned} g_{p}(z) := g_{0}\exp (-a_0z), \end{aligned}$$
where \(g_{0} < 1\) is the probability of a seed on the surface of the soil germinating (which is dimensionless) and \(a_{0}\) models the loss in germination probability that occurs through an incremental increase in seed depth (Mohler 1993). The parameter \(a_{0}\) has the dimensions of \((depth)^{-1}\).
The depth distribution of new seeds is assumed to be (truncated) exponentially distributed with mean \(c_{0} << D\), so
$$\begin{aligned} J(z) := c_0^{-1}\exp (-z/c_0) \end{aligned}$$
for \(x < D\) and \(J(D) = 1 - \exp (- D/c_0) > 0\) but \(\ll 1\). This assumption takes into account that most seeds that are produced will end up near the surface of the soil, while few will end up in deeper parts of the soil. The dimension of \(c_{0}\) is \(depth\).
We assume finally that recruitment follows a Holling Type II functional form (Holling 1959). A derivation of an analogous relationship for a general plant population is available in Eager et al. (2012), which takes into account that plants compete for a finite number of available microsites (“predation of space”). The number of plants that result from \(z \; seeds(area)^{-1}\) is thus
$$\begin{aligned} f(z) := \frac{\alpha z}{\beta + z}. \end{aligned}$$
As summarized in Eager et al. (2012), the parameter \(\alpha \) is the maximum density of adult plants that can grow in a given area, with dimension \(plants(area)^{-1}\). The parameter \(\beta \) (with dimension \(seeds(area)^{-1}\)) is the half saturation constant. Recall that the dimensions of \(f\) are \(plants(area)^{-1}\).
We performed a simulation study to see how the disturbance parameters \(h\) and \(\rho \) affect (1) the long-term mean of the total seed bank population \(||n(\cdot , t)||_{L^1}\), and (2) the rare growth rate \(\lambda \). The results from (1) show how the long-term abundance of seeds in the seed bank changes with respect to changes in disturbance regimes. The results from (2) show how these changes elicit changes in the likelihood of population extinction. In Eager et al. (2013a) we showed that the qualitative ways in which the disturbance parameters change the long-term mean of the seed bank and the likelihood of extinction are relatively insensitive to changes in the other population parameters, except for the maximum possible seed survival probability \(s_0\), which we will discuss momentarily. To approximate the long-term mean of the seed bank population we simulated the population out to \(t = 10000\) 500 times and took the mean of the \(L^{1}\) norms of the \(500 n(\cdot , 10000)\) populations. To approximate \(\lambda \) we simulated the population out until \(t = 150\) 500 times and took the mean of the \(500\)\(\lambda \) values at \(t = 150\). The results are summarized in Fig. 1. We used a numerical integration technique described in Ellner and Rees (2006), and the parameter values \(s_0 = 0.5\) (Fig. 1a, c) \(s_0 = 0.95\) (Fig. 1b, d)\(, g_0 = 0.95, a_0 = b_0= 10, c_0 = 100, \alpha = 40, \beta = 50, \kappa = 150, \epsilon _0 = 0.001\) and \(\sigma = 0.002\) (Fig. 1a, b, c, d), with approximating matrices of dimension \(N = 30\).
Fig. 1

The relationship between the mean of the long-term total seed bank population \(||n(\cdot ,t)||_{L^1}\) (a, b), approximations of the rare growth rate \(\lambda \) (c, d) and the probability of disturbance \(h\) for various mean depths of disturbances \(\rho \). The dashed line in (c) and (d) represent the extinction threshold, i.e., if \(\lambda < 1\) the population goes extinct as \(t \rightarrow \infty \) with probability 1 and if \(\lambda > 1\) it persists as \(t \rightarrow \infty \) with probability 1. The parameter values used are \(s_0 = 0.5\) ((a) and (c)) \(s_0 = 0.95\) ((b) and (d))\(, g_0 = 0.95, a_0 = b_0= 10, c_0 = 100, \alpha = 40, \beta = 50\) and \(\kappa = 150\) (Color figure online)

Note that both the long-term mean of the seed bank population \(||n(\cdot , t)||_{L^1}\) and the rare growth rate \(\lambda \) are increasing functions of the probability of disturbance \(h\), while the long-term mean of the seed bank population \(||n(\cdot , \cdot )||_{L^1}\) can exhibit a nonmonotone relationship with \(\rho \) if maximum seed survival \(s_0\) is small (Fig. 1a). Also note that \(\lambda \) decreases with increasing \(\rho \) (Fig. 1c, d). These results suggest that having a larger mean depth of disturbance \(\rho \) is beneficial when the population is well above extinction, but having a lower \(\rho \) near extinction levels is actually more beneficial than having a high \(\rho \), especially when the survival of seeds deep in the seed bank is low. When \(s_0\) is small we see a decreasing relationship between long-term seed bank abundance and \(\rho \) because, as \(\rho \) increases, more newly created seeds are being brought from the surface of the soil (where germination is high) to deeper layers where germination is rare and seeds are likely to die before being brought back to surface layers via disturbance. This is in contrast to when \(s_{0}\) is large, as deep soil layers produce a storage effect, i.e. they provide a way of buffering against the effects of years without disturbance, and thus we see an increasing relationship between long-term population abundance and \(\rho \). This phenomenon was also seen in simulation studies using the model in Eager et al. (2013a).

6 Extensions

There are some extensions that could be made to the model studied in this paper to make it more biologically realistic. One possible extension is to consider sequences of random variables \(\{\omega _t\}_{t=0}^{\infty }\) that are not roughly \(iid\). There has been an increasing recognition among theoretical, empirical, and mathematical ecologists of the importance of long-term correlations in environmental conditions over time (Heino et al. 2000; Vasseur and Yodzis 2004; Benaïm and Schreiber 2009; Lögdberg and Wennergren 2012; Mustin et al. 2013). Because we are considering only natural disturbances in this setting, the same can be said for seemingly randomly disturbances, which are influenced by a seemingly endless array of vectors that are influenced by environmental conditions themselves. Benaïm and Schreiber (2009) successfully extended the asymptotic results of Chession (1982), Ellner (1984) and Hardin et al. (1988) for finite-dimensional population models with periodic, quasi-periodic, Markovian, and auto-regressive moving average environments. Weakening our roughly \(iid\) assumption on the sequence \(\{\omega _t\}_{t=0}^{\infty }\) in our infinite-dimensional setting may force us to significantly alter our approach to showing that the population in (1) converges in distribution, as one of the main ingredients in our proof was our ability to shift our attention from the sequence of random variables \(\{\omega _t\}_{t=0}^{\infty }\) to the sequence of random variables \(\{\hat{\omega }_{\hat{t}}\}_{t=0}^{\infty }\) sharing the same \(iid\) statistical properties in relatively easy way.

Another possible extension is to build upon the ideas in Eager et al. (2013b) and structure the seeds in the seed bank with respect to their age, as well as their depth. In this setting, the seed bank population will also be a function of age \(a \in \{1,2, \ldots , m\}\) as well as depth. In this case, germination and survival would be a function of both age and depth, leading to the following model
$$\begin{aligned} n(z, 1, t + 1)&= \mathop \int \limits _0^D s(z, 0)(1-g(z, 0,\omega ))K(z,y,\omega )cJ(y)p(t)\, \mathrm{d}y\nonumber \\ n(z, a, t + 1)&= \mathop \int \limits _0^D s(z, a - 1)(1-g(z, a - 1,\omega ))K(z,y,\omega )n(y, a -1, t)\, \mathrm{d}y,\nonumber \\&a = 2, 3,\ldots , m -1 \nonumber \\ n(z, m, t + 1)&= \mathop \int \limits _0^D s(z, m - 1)(1-g(z, m - 1,\omega ))K(z,y,\omega )(n(y, m -1, t) \nonumber \\&+\, n(y, m, t))\, \mathrm{d}y \nonumber \\ p(t + 1)&= f\left( \left( \sum _{a=1}^m\mathop \int \limits _0^D \mathop \int \limits _0^D g(z, a,\omega )K(z,y,\omega )n(y,t)\, \mathrm{d}y\, \mathrm{d}z\right) \right. \nonumber \\&\left. + \mathop \int \limits _0^D\mathop \int \limits _0^D g(z, 0,\omega )K(z,y,\omega ) cJ(y)p(t))\, \mathrm{d}y\, \mathrm{d}z\right) . \end{aligned}$$
The analysis of the model (21) will also potentially need a substantial modification from that in Theorem 1, as properties (H2 (c)) and (H3) are not satisfied. To see this note that if there is not a disturbance at time \(t\) then the density of seeds of age \(1\) at time \(t + 1\) will be zero (similarly for seeds of age \(2\) at time \(t + 2\), etc). Thus, when it is possible to have a time-step with no disturbance the seed bank population \(n(\cdot ,\cdot ,t)\) will be on the boundary of \((C^{+}[0,D])^m\) for many time-steps \(t\), and thus no such constants \(m\) and \(\eta \) from properties (H2 (c)) and (H3), respectively, exist to bound the population from below. Some simulations studies by the authors suggest that the conclusions of Theorem 1 still hold for (21), but the analysis will have to wait for another manuscript.

Finally, while the assumptions that \(T < \infty \) and \(\sigma > 0\) are ecologically justifiable in this setting, for the purposes of applying the techniques in this paper to other ecological settings, it may be important to explore what happens when one allows \(T \rightarrow \infty \) and/or \(\sigma \rightarrow 0\). Relaxing these assumptions causes the the upper bound in (2), the lower bounds in (13) and (14) and the continuity condition (18) to no longer apply, implying that (H3) and (H5) are no longer true, meaning the population is not guaranteed to be bounded from below during long disturbance-free periods. Since (H3) and (H5) are main ingredients in the proof of Theorem 4.2 in Hardin et al. (1988—see equation (11) and (12) in that paper), if one were to relax the assumptions on \(T\) and \(\sigma \), we would not necessarily expect analogous results.

7 Discussion

In this paper, we develop and analyze a coupled plant-seed bank model for a general disturbance specialist plant population and its seed bank. We assume that the seed bank’s life-history parameters (survival and germination) depend on a seed’s depth in the seed bank, which provides us with the motivation to use depth as the model’s (continuous) stage variable. The main biological feature of disturbance specialists is that seeds do not germinate in the absence of disturbances, which causes part of the population (plants) to rest on the boundary of its positive cone for many time-steps. However, in the absence of disturbance, the model is entirely deterministic. We use this insight to modify a proof of Hardin et al. (1988), showing that the plant-seed bank population converges to a stationary distribution, independent of (non-zero) initial population. We show there is a dichotomy in the long-term fate of the population, as it either goes extinct with probability \(1\) or persists with probability \(1\) and is determined by the rare growth rate \(\lambda = \lim _{t\rightarrow \infty }||A(\omega _t) \circ A(\omega _{t-1}) \circ \cdots \circ A(\omega _0)1\!\!1||_{\infty }^{1/t}\). We illustrate these results with a simulation study motivated by wild sunflower (Helianthus annuus) populations.

The biological implication of this work is that we have isolated a tradeoff between missing germination opportunities and a storage effect, which is illustrated in Fig. 1. When disturbances are rare (small \(h\)), but deep (high \(\rho \)), there is a high probability of seeds being moved to deeper depths, where the germination rate is very low. If seed survival in these deep layers is low (small \(s_0\), see Fig. 1a, c), then a seed has a high chance of dying before it can reach the surface of the soil again and germinate. Thus, in this case shallow disturbances are more advantageous for seeds because they allow them to stay close to the surface and not miss the opportunity to germinate, resulting in higher population numbers and a lower extinction risk for smaller \(\rho \). Conversely, when seed survival in deep layers is high (high \(s_0\), see Fig. 1b, d), seeds have a high chance of surviving until the next time they are brought to the surface of the soil via disturbance, even though these events can be infrequent. Thus, in this case deeper disturbances are beneficial for plants because they move seeds to safe sites and buffer the population against the negative impact of a long period without a germination opportunity (i.e. no disturbance), causing the increasing relationship between long-term population size and \(\rho \) seen in Fig. 1b, d. If disturbance events are frequent (high \(h\)) there are always a relatively large number of seeds on the surface ready to germinate, and thus even a weak storage effect (like \(s_0 = 0.5\) in Fig. 1a, c) is beneficial and thus larger \(\rho \) always increases the long-term population abundance in this case.



We would like to thank Professors Diana Pilson and Steven Dunbar for useful discussions about this work, and the two anonymous referees for suggestions which greatly improved the paper.


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Copyright information

© Society for Mathematical Biology 2014

Authors and Affiliations

  • Eric Alan Eager
    • 1
  • Richard Rebarber
    • 2
  • Brigitte Tenhumberg
    • 3
  1. 1.University of Wisconsin - La CrosseLa CrosseUSA
  2. 2.Department of MathematicsUniversity of Nebraska–LincolnLincolnUSA
  3. 3.School of Biological SciencesUniversity of Nebraska–LincolnLincolnUSA

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