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

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## Abstract

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.

### Keywords

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).

*t*th 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.

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 \(plants\).

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 \(w_t > 0\)).

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 \(C[0,D]\).

### 3.2 Reformulation of the Model (1)

*t*th 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

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.

- (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 \(t\).

- (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 \(\overline{K}\), andfor 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 \(\omega \).$$\begin{aligned} \overline{s}||\mathop \int \limits _0^D K(\cdot , y, \omega _2)x(y) \mathrm{d}y||_{\infty } \le ||x(\cdot )||_{\infty } \end{aligned}$$(12)
- (A3)
The function \(K \in C^1([0,D]^2 \times [\sigma , D])\)

- (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 \(z \in [0,D]\).

- (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 \(f(a)/a < f(b)/b\).

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).

- (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 \(x = 0 \in C[0,D]^{+}\).

- (H2)There exists some \(M > 0\) such that for all \(x \in C[0,D]^{+}\) and \(\omega \in \varOmega \),
- (a)
\(||H(\omega , x)||_{\infty } \le M\) whenever \(||x||_{\infty } \le M.\)

- (b)There is some time-step \(t\) (which depends on \(x_0\)) \(\in \mathbb {N}\) such that$$\begin{aligned} ||H(\omega _t, x_t)\circ H(\omega _{t-1}, x_{t-1})\circ \cdots \circ H(\omega _0, x_0)||_{\infty } < M. \end{aligned}$$
- (c)There is some \(m > 0\) such thatfor all \(\omega \in \varOmega \).$$\begin{aligned} H(\omega ,M1\!\!1) \ge m1\!\!1, \end{aligned}$$

- (a)
- (H3)For \(M\) as in (H2) letThere 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$$\begin{aligned} B_M := \{x \in C[0,D]^{+}: ||x||_{\infty } \le M \}. \end{aligned}$$for all \(\omega \in \varOmega .\)$$\begin{aligned} H(\omega ,B_M) \subset \{x \in C[0,D]^{+}: x \ge \eta ||x||_{\infty }1\!\!1\} := m_{\eta } \end{aligned}$$
- (H4)There exists some \(h >0\) such thatfor all \(\omega \in \varOmega \) and \(x \in C[0,D]^{+}\).$$\begin{aligned} ||H(\omega ,x)||_{\infty } \le h ||x||_{\infty }, \end{aligned}$$
- (H5)For each \(M_0 > 0\) there exists a function \(\psi :(0,1] \rightarrow (0,1]\) such that \(\psi (v) > v\) for \(v \in (0,1)\) andfor all \(\omega \in \varOmega \) and \(x \in C[0,D]^{+}\) for which \(x \in [M_0,M]\), where \(M\) is from (H2).$$\begin{aligned} \psi (v)H(\omega ,x) \le H(\omega ,vx), \end{aligned}$$
- (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 \(0 \in C[0,D]^{+}\).

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

## 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 \(\{H(\omega ,\cdot )\}_{\omega \in \varOmega }\).

**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 \(\mu ^{*}(\{0\}) = 1\).

*Proof*

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 (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.

### 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 \(\rho \)).

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.

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.

## Notes

### Acknowledgments

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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