Effects of Patch–Matrix Composition and Individual Movement Response on Population Persistence at the Patch Level

Abstract

Fragmentation creates landscape-level spatial heterogeneity which in turn influences population dynamics of the resident species. This often leads to declines in abundance of the species due to increased susceptibility to edge effects between the remnant habitat patches and the lower quality “matrix” surrounding these focal patches. In this paper, we formalize a framework to facilitate the connection between small-scale movement and patch-level predictions of persistence through a mechanistic model based on reaction–diffusion equations. The model is capable of incorporating essential information about edge-mediated effects such as patch preference, movement behavior, and matrix-induced mortality. We mathematically analyze the model’s predictions of persistence with a general logistic-type growth term and explore their sensitivity to demographic attributes in both the patch and matrix, as well as patch size and geometry. Also, we provide bounds on demographic attributes and patch size in order for the model to predict persistence of a species in a given patch based on assumptions on the patch/matrix interface. Finally, we illustrate the utility of this framework with a well-studied planthopper species (Prokelisia crocea) living in a highly fragmented landscape. Using experimentally derived data from various sources to parameterize the model, we show that, qualitatively, the model results are in accord with experimental predictions regarding minimum patch size of P. crocea. Through application of a sensitivity analysis to the model, we also suggest a ranking of the most important model parameters based on which parameter will cause the largest output variance.

Appendices

Sensitivity Analysis Methodology

In this subsection, we will briefly describe the sensitivity analysis methodology which was applied to the model. Following Saltelli et al. (2008, 2010), we generate a ten-dimensional list of quasi-random parameter values, i.e., tuples of the form $(x_1^0, x_2^0, x_3^0, x_4^0, x_5^0, x_1^1, x_2^1, x_3^1, x_4^1, x_5^1)$, of length N based on the Sobol quasi-random sequence. Two matrices $\mathbf {A}$ and $\mathbf {B}$ of size (N, 5) are then constructed using the first half of the N tuples for the rows of $\mathbf {A}$ and the remainder for the rows of $\mathbf {B}$. Thus, a row of the matrix $\mathbf {A}$ or $\mathbf {B}$ will contain values for these five parameters. A third matrix $\mathbf {A_B^{s_1, s_2, \ldots s_i}}$ is then constructed using all the columns of $\mathbf {A}$ except columns $s_1, s_2, \ldots s_i$ are taken from the matrix $\mathbf {B}$, where $s_1, s_2, \ldots s_i, i \in \{1, 2, 3, 4, 5\}$. For example, $\mathbf {A_B^{2}}$ is matrix A where the second column is taken from $\mathbf {B}$ and $\mathbf {A_B^{2, 3}}$ is matrix A where the second and third columns are taken from $\mathbf {B}$.

We now define the function $h(\mathbf {x})$ as the output of the model (either minimum patch size, minimum intrinsic growth rate, or maximum patch diffusion rate) given the five parameters $\mathbf {x} = (x_1, x_2, x_3, x_4, x_5)$ and a fixed patch geometry in $\varOmega _0$. We then compute the model output for all input values in the three matrices, $\mathbf {A}, \mathbf {B},$ and $\mathbf {A_B^{s_1, s_2, \ldots , s_i}}$ giving the N-dimensional vectors $h(\mathbf {A})$, $h(\mathbf {B})$, and $h(\mathbf {A_B^{s_1, s_2, \ldots , s_i}})$. Using these matrices, we can generate an estimate of the first-order interaction (main effect) of $x_i$

$$\begin{aligned}&S_{x_i} = \frac{\frac{1}{N} \sum _{m=1}^N \left[ h(\mathbf {A}_m) \left( h\left( (\mathbf {A_B^i})_m\right) - h(\mathbf {B}_m) \right) \right] }{\frac{1}{2N - 1} \sum _{m = 1}^N \left[ (h(\mathbf {A}_m) - h_0)^2 + (h(\mathbf {B}_m) - h_0)^2 \right] } \end{aligned}$$

and total effect index of $x_i$

$$\begin{aligned}&S_{T_{x_i}} = \frac{\frac{1}{2N} \sum _{m=1}^N \left( h(\mathbf {B}_m) - h\left( (\mathbf { A_B^i)}_m\right) \right) ^2}{\frac{1}{2N - 1} \sum _{m = 1}^N \left[ (h(\mathbf {A}_m) - h_0)^2 + (h(\mathbf {B}_j) - h_0)^2 \right] }, \end{aligned}$$

where $i \in \{1, 2, 3, 4, 5\}$, $h_0 = \frac{1}{2N - 1} \sum _{m = 1}^N \left[ h(\mathbf {A}_m) + h(\mathbf {B}_m) \right]$, and $\mathbf {A}_m, \mathbf {B}_m$ represent the m-th rows of the matrices $\mathbf {A}$ and $\mathbf {B}$, respectively.

Since the modeling framework is presented in a general form in order to accommodate as many species as possible, no species-specific parameter ranges were chosen. Instead, all the parameters’ ranges were set to [0.01, 100], with the exception of the probability of remaining in the patch upon reaching the patch/matrix interface, $\alpha$, whose range was chosen as [0.01, 0.99]. An algorithm was implemented in Mathematica (Wolfram Research Inc., version 11.3), (1) to compute estimates of these sensitivity indices. In the algorithm, N was initially set to 250 and the main effect indices were calculated iteratively as N was incremented by 250 each time. This process continued until the norm of the difference between the vectors of successive main effect indices was within our predetermined goal of 0.0005, indicating convergence of the estimated indices to the actual ones. The final value of N was then used to compute estimates for the total effect index. A simple geometry was chosen for $\varOmega _0$ in that it was only considered as a disk in two spatial dimensions. As changes in patch geometry affect the predictions of the model, an extension of our results could include preforming a sensitivity analysis on the model with patch geometry counted and varied as a parameter in the analysis. This is, however, out of the scope of this work.

Statement and Proof of Lemma 1

Lemma 1

Let $\lambda _1(\varOmega _0, \beta )$ be the principal eigenvalue of (14) with corresponding eigenfunction, $\phi$, which is chosen such that $\phi > 0;\ x \in \overline{\varOmega }_0$ and $||\phi ||_\infty = 1$. Then, we have the following:

• (a) $\lambda _1(\varOmega _0, 0) = 0$, $\lambda _1(\varOmega , \beta )$ is a strictly increasing function of $\beta$, and $\lambda _1(\varOmega _0, \beta ) \rightarrow \lambda _1^0(\varOmega _0)$ as $\beta \rightarrow \infty$, where $\lambda _1^0(\varOmega _0)$ is the principal eigenvalue of Laplace’s equation with Dirichlet boundary conditions ($u = 0;\ x \in \partial \varOmega _0$).

• (b) $\lambda _1(\varOmega _0, \beta )$ is a differentiable function of $\beta$.

• (c) $\lambda _1(\varOmega _0, \beta )$ is a concave function of $\beta$.

The proof of (a) and (b) is standard, see Cantrell and Cosner (2003). The proof of (c) is as follows. For brevity, we denote $\lambda _1(\beta ) = \lambda _1(\varOmega _0, \beta )$. We begin by differentiating (14) with respect to $\beta$ yielding

$$\begin{aligned}&-\varDelta \phi '(\beta ) = \lambda _1'(\beta ) \phi (\beta ) + \lambda _1(\beta )\phi '(\beta );\ x \in \varOmega _0 \nonumber \\&\quad \frac{\partial \phi '(\beta )}{\partial \eta } + \phi (\beta ) + \beta \phi '(\beta ) = 0;\ x \in \partial \varOmega _0, \end{aligned}$$

(26)

where $'$ denotes differentiation with respect to $\beta$. Next, we calculate $\lambda _1'(\beta )$ for any $\beta > 0$. By Green’s second identity, we have:

$$\begin{aligned} \int _{\varOmega _0} \left[ \left( -\varDelta \phi (\beta ) \right) \phi '(\beta ) + \phi (\beta ) \left( \varDelta \phi '(\beta ) \right) \right] \hbox {d}x= & {} \int _{\partial \varOmega _0} \frac{-\partial \phi (\beta )}{\partial \eta } \phi '(\beta )\nonumber \\&+ \phi (\beta ) \frac{\partial \phi '(\beta )}{\partial \eta } \hbox {d}s \end{aligned}$$

(27)

But, we also have that

$$\begin{aligned}&\int _{\varOmega _0} \left[ \left( -\varDelta \phi (\beta ) \right) \phi '(\beta ) + \phi (\beta ) \left( \varDelta \phi '(\beta ) \right) \right] \hbox {d}x\nonumber \\&\quad = \int _{\varOmega _0} \lambda _1(\beta ) \phi (\beta ) \phi '(\beta ) - \phi (\beta )\lambda _1'(\beta )\phi (\beta ) - \lambda _1(\beta ) \phi (\beta ) \phi '(\beta )\hbox {d}x \nonumber \\&\quad = -\lambda _1'(\beta ) \int _{\varOmega _0} \left[ \phi ^2(\beta ) \right] \hbox {d}x \end{aligned}$$

(28)

and

$$\begin{aligned}&\int _{\partial \varOmega _0} \frac{-\partial \phi (\beta )}{\partial \eta } \phi '(\beta ) + \phi (\beta ) \frac{\partial \phi '(\beta )}{\partial \eta } \hbox {d}s \nonumber \\&\quad = \int _{\partial \varOmega _0} \left[ \beta \phi (\beta ) \phi '(\beta ) - \phi ^2(\beta ) -\beta \phi (\beta ) \phi '(\beta ) \right] \hbox {d}s \nonumber \\&\quad = -\int _{\partial \varOmega _0} \phi ^2(\beta ) \hbox {d}s. \end{aligned}$$

(29)

Combining (28) and (29) gives,

$$\begin{aligned} \lambda _1'(\beta ) = \frac{\int _{\partial \varOmega _0} \phi ^2(\beta ) \hbox {d}s}{\int _{\varOmega _0} \phi ^2(\beta ) \hbox {d}x} > 0;\ \beta \ge 0. \end{aligned}$$

(30)

Now, by Green’s first identity and (14), we have

$$\begin{aligned} \int _{\varOmega _0} \left[ \left( -\varDelta \phi (\beta ) \right) \phi (\beta ) \right] \hbox {d}x= & {} \int _{\varOmega _0} \left| \nabla \phi (\beta )\right| ^2\hbox {d}x - \int _{\partial \varOmega _0} \frac{\partial \phi (\beta )}{\partial \eta } \phi (\beta ) \hbox {d}s\nonumber \\= & {} \int _{\varOmega _0} \left| \nabla \phi (\beta )\right| ^2 \hbox {d}x + \beta \int _{\partial \varOmega _0} \phi ^2(\beta ) \hbox {d}s. \end{aligned}$$

(31)

Also, from (14) we have that

$$\begin{aligned} \int _{\varOmega _0} \left[ \left( -\varDelta \phi (\beta ) \right) \phi (\beta ) \right] \hbox {d}x= & {} \lambda _1(\beta ) \int _{\varOmega _0} \phi ^2(\beta ) \hbox {d}x. \end{aligned}$$

(32)

Combining (31) and (32) and solving for $\int _{\partial \varOmega _0} \phi ^2(\beta ) \hbox {d}s$ yields

$$\begin{aligned} \int _{\partial \varOmega _0} \phi ^2(\beta ) \hbox {d}s= & {} \frac{1}{\beta } \lambda _1(\beta ) \int _{\varOmega _0} \phi ^2(\beta ) \hbox {d}x - \frac{1}{\beta } \int _{\varOmega _0} \left| \nabla \phi (\beta )\right| ^2 \hbox {d}x. \end{aligned}$$

(33)

Thus, if we combine (30) with (33), then we have

$$\begin{aligned} \lambda _1'(\beta )= & {} \frac{1}{\beta } \lambda _1(\beta ) - \frac{1}{\beta \int _{\varOmega _0} \phi ^2(\beta ) \hbox {d}x} \int _{\varOmega _0} \left| \nabla \phi (\beta )\right| ^2 \hbox {d}x;\ \beta > 0. \end{aligned}$$

(34)

Hence,

$$\begin{aligned} \lambda _1'(\beta )\le & {} \frac{\lambda _1(\beta )}{\beta };\ \beta > 0, \end{aligned}$$

(35)

which proves that $\lambda _1(\beta )$ is a concave function for $\beta > 0$. $\square$

Special Case of a Disk-Shaped Patch in Two Dimensions

In this subsection, we present a derivation and mathematical analysis of a mechanistically correct model (at least in the sense of steady states and their stability properties) in the special case of a disk-shaped patch in two dimensions with radius $\ell > 0$ and patch population density denoted by $v(t,\rho )$, namely

$$\begin{aligned}&v_t = D \left( v_{\rho \rho } + \frac{1}{\rho } v_\rho \right) + rvf(v);\ t> 0, \rho \in (0, \ell ) \nonumber \\&v(0, \rho ) = v_0(\rho );\ \rho \in (0,\ell ) \nonumber \\&v_\rho = 0;\ t> 0,\ \rho = 0 \nonumber \\&D v_\rho + \frac{\sqrt{S_0 D_0}}{\kappa } H\left( \sqrt{\frac{S_0}{D_0}} \ell \right) v = 0;\ t > 0,\ \rho = \ell \end{aligned}$$

(36)

where

$$\begin{aligned} H(s) = \frac{K_1(s)}{K_0(s)}, \end{aligned}$$

(37)

and $K_0, K_1$ are modified Bessel functions of the second kind with all the parameters defined as in Sect. 2.

1.1 Derivation

Following the modeling setup as in Sect. 2, we assume that the patch is disk-shaped with radius $\ell > 0$, i.e., $\varOmega = \left\{ x \in \mathbb {R}^2\ |\ |x| < \ell \right\}$, with u(t, x) representing the density in $\varOmega$, and the matrix is the region exterior to $\varOmega$, i.e., $\varOmega _e = \left\{ x \in \mathbb {R}^2\ |\ |x| \ge \ell \right\}$, with w(t, x) representing the density in the matrix. Assuming a growth law similar to (2) in the matrix and the same patch/matrix interface assumptions as in Sect. 2.1, we have the following system to describe the population dynamics of an organism in this patch/matrix system:

$$\begin{aligned}&u_t = D \varDelta u + ruf(u);\ t> 0, x \in \varOmega \nonumber \\&w_t = D_0 \varDelta w -S_0 w;\ t> 0, x \in \varOmega _e \nonumber \\&u(0, x) = u_0(x);\ x \in \varOmega \nonumber \\&w(0, x) = 0;\ x \in \varOmega _e \nonumber \\&D \frac{\partial u}{\partial \eta } = D_0 \frac{\partial w}{\partial \eta _0};\ t> 0, |x| = \ell \nonumber \\&u(t,x) = \kappa w(t,x);\ t> 0, |x| = \ell \nonumber \\&w(t,x) \rightarrow 0;\ t > 0, \text{ as } |x| \rightarrow \infty . \end{aligned}$$

(38)

Making use of the rotational symmetry of the patch/matrix system, we convert the system $\left( u(t,x), w(t,x)\right)$ to $\left( v(t,\rho ), \omega (t,\rho )\right)$ yielding

$$\begin{aligned}&v_t = D \left( v_{\rho \rho } + \frac{1}{\rho } v_\rho \right) + rvf(v);\ t> 0, \rho \in (0, \ell ) \nonumber \\&\omega _t = D_0 \left( \omega _{\rho } + \frac{1}{\rho } \omega _\rho \right) - S_0 \omega ;\ t> 0, \rho \in (\ell , \infty )\nonumber \\&v(0, \rho ) = v_0(\rho );\ \rho \in (0,\ell ) \nonumber \\&\omega (0, \rho ) = 0;\ \rho \in (\ell , \infty ) \nonumber \\&D v_\rho = D_0 \omega _\rho ;\ t> 0,\ \rho = \ell \nonumber \\&v(t,\ell ) = \kappa \omega (t,\ell );\ t> 0 \nonumber \\&v_\rho = 0;\ t> 0,\ \rho = 0 \nonumber \\&\omega (t,\rho ) \rightarrow 0;\ t > 0, \text{ as } \rho \rightarrow \infty . \end{aligned}$$

(39)

We now make the assumption that the population density is at a stationary state in the matrix which must be of the form $\omega (\rho ) = C_1 K_0\left( \sqrt{\frac{S_0}{D_0}} \rho \right)$ for $\rho \ge \ell$ [see, for example, Skellam (1951)]. Notice that $\omega '(\rho ) = -C_1 \sqrt{\frac{S_0}{D_0}} K_1\left( \sqrt{\frac{S_0}{D_0}} \rho \right)$. Thus, applying the interface conditions at $\rho = \ell$ in (39) to this solution yields

$$\begin{aligned} D v_\rho + \frac{\sqrt{S_0 D_0}}{\kappa } \frac{K_1\left( \sqrt{\frac{S_0}{D_0}} \ell \right) }{K_0\left( \sqrt{\frac{S_0}{D_0}} \ell \right) } v = 0;\ t > 0,\ \rho = \ell . \end{aligned}$$

(40)

With this Robin boundary condition, it is now possible to consider the problem only inside the patch via the dynamical problem, (36). Although the nonstationary solutions of (36) are not equivalent to those of the original patch/matrix system in (38), the argument given in Potapov and Lewis (2004) ensures that the stationary solutions of (36) and their stability properties are equivalent to the ones in the original system.

Applying the scaling

$$ \begin{aligned} \tilde{\rho } = \frac{\rho }{\ell }\ \& \ \tilde{t} = r t, \end{aligned}$$

(41)

and dropping the tilde, (36) becomes:

$$\begin{aligned}&v_t = \frac{1}{\lambda } \left( v_{\rho \rho } + \frac{1}{\rho } v_\rho \right) + vf(v);\ t> 0, \rho \in (0, 1) \nonumber \\&v(0, \rho ) = v_0(\rho );\ \rho \in (0, 1) \nonumber \\&v_\rho = 0;\ t> 0,\ \rho = 0 \nonumber \\&v_\rho + \frac{\ell \sqrt{S_0 D_0}}{\kappa D} H\left( \sqrt{\frac{S_0}{D_0}} \ell \right) v = 0;\ t > 0,\ \rho = 1 \end{aligned}$$

(42)

where $\lambda = \frac{r \ell ^2}{D}$ is unitless. In Table 13, we enumerate the different possibilities for the boundary condition of (42) as the different scenarios and patch attributes are changed. We then write the multiparameter model (42) in such a way as to ensure that the original parameters are split into three unitless composite parameters, namely $\lambda , \gamma _1,$ and $\gamma _2$, in such a way that the parameter in question (r, D,  or $\ell$) occurs only in $\lambda$. This process gives several different forms of the boundary condition in (42) as listed in Table 13. We note that under the assumption of $D = D_0$ in CD and Type III DD interface scenarios, the boundary conditions for these two cases are different than that of Type I DD. Recall that the CD scenario can be considered as a special case of Type III DD in which $\alpha = \frac{1}{2}$.

Table 13 Boundary condition possibilities for (42)

As in Sect. 2, all twelve parameters of interest and interface condition pairs can be treated mathematically via study of the multiparameter problem:

$$\begin{aligned}&v_t = \frac{1}{\lambda } \left( v_{\rho \rho } + \frac{1}{\rho } v_\rho \right) + vf(v);\ t> 0,\ \rho \in (0, 1) \nonumber \\&v(0, \rho ) = v_0(\rho );\ \rho \in (0, 1) \nonumber \\&v_\rho = 0;\ t> 0,\ \rho = 0 \nonumber \\&v_\rho + \lambda ^{\mu _1} \gamma _1 H(\lambda ^{\mu _2} \gamma _2) v = 0;\ t > 0,\ \rho = 1 \end{aligned}$$

(43)

where $\mu _1, \mu _2 = 0, \frac{1}{2}$ and the meaning of the unitless parameters $\gamma _1, \gamma _2$ will depend on the parameter of interest and interface type.

1.2 Mathematical Analysis of (43)

As in Sect. 3, the dynamics of (43) are almost completely determined by its steady states, i.e., solutions of

$$\begin{aligned}&-\left( v'' + \frac{1}{\rho } v_\rho \right) = \lambda vf(v);\ \rho \in (0, 1) \nonumber \\&v'(0) = 0\nonumber \\&v'(1) + \lambda ^{\mu _1} \gamma _1 H(\lambda ^{\mu _2} \gamma _2) v(1) = 0. \end{aligned}$$

(44)

Given a solution of (44), v, its local stability properties can be determined by examining the sign of the principal eigenvalue, $\sigma _1 = \sigma _1(\varOmega _0, \lambda , \gamma _1, \gamma _2, \mu _1, \mu _2, v)$, of the linearized eigenvalue problem associated with (44):

$$\begin{aligned}&-\left( \phi '' + \frac{1}{\rho } \phi ' \right) -\lambda \left[ f(v) + v f'(v)\right] \phi = \sigma \phi ;\rho \in (0, 1) \nonumber \\&\phi '(0) = 0\nonumber \\&\phi '(1) + \lambda ^{\mu _1} \gamma _1 H(\lambda ^{\mu _2} \gamma _2) \phi (1) = 0 \end{aligned}$$

(45)

with $\varOmega _0 = (0, 1)$ and corresponding eigenfunction, $\phi$, which can be chosen such that $\phi > 0;\ x \in [0, 1]$ and $||\phi ||_\infty = 1$.

As in Sect. 3, model predictions of persistence can be determined by studying the stability of the trivial steady state, $v \equiv 0$, via consideration of the sign of $\sigma _1(\varOmega _0, \lambda , \gamma _1, \gamma _2, \mu _1, \mu _2, 0)$. In the case of a reaction term satisfying the logistic-type assumptions, (F1) and (F2), Theorem 1 allows an exact description of the global dynamics of (43) based solely on the sign of $\sigma _1(\varOmega _0, \lambda , \gamma _1, \gamma _2, \mu _1, \mu _2, v)$. In the following analysis, we will explicitly describe the relationship between a given parameter of interest ($r, D, \text{ or } \ell$) and the persistence of the population under each of the four interface scenarios given in Table 1 via comparison of the linearized eigenvalue problem for the trivial steady state, (45), with the eigenvalue problem (14). Uniqueness of the principal eigenvalue again implies that:

$$\begin{aligned}&\lambda _1(\varOmega _0, \beta ) = \sigma _1(\varOmega _0, \lambda , \gamma _1, \gamma _2, \mu _1, \mu _2, 0) + \lambda \end{aligned}$$

(46)

$$\begin{aligned}&\beta = \lambda ^{\mu _1} \gamma _1 H(\lambda ^{\mu _2} \gamma _2). \end{aligned}$$

(47)

In the case of $\mu _2 = 0$ and $\mu _1 = 0$ or $\frac{1}{2}$, the mathematical analysis of (43) exactly follows that of Sect. 4 with $\gamma = \gamma _1 H(\gamma _2)$. The only remaining case is $\mu _2 = \mu _1 = \frac{1}{2}$, for which (46) & (47) become:

$$\begin{aligned} \sigma _1(\varOmega _0, \lambda , \gamma _1, \gamma _2, 0.5, 0.5, 0) = \lambda _1(\varOmega _0, \sqrt{\lambda } \gamma _1 H(\sqrt{\lambda } \gamma _2)) - \lambda \end{aligned}$$

(48)

Theorem 4 connects the sign of $\sigma _1(\varOmega _0, \lambda , \gamma _1, \gamma _2, 0.5, 0.5, 0)$ to ranges of parameter space for $\lambda$, $\gamma _1$, and $\gamma _2$.

Theorem 4

Let $\gamma _1, \gamma _2 > 0$. Then, we have the following:

• (a) there exists a $\lambda _3^*(\varOmega _0, \gamma _1, \gamma _2) > 0$ such that

  • (1) if $\lambda \le \lambda _3^*(\varOmega _0, \gamma _1, \gamma _2)$, then $\sigma _1(\varOmega _0, \lambda , \gamma _1, \gamma _2, 0.5, 0.5, 0) \ge 0$

  • (2) if $\lambda > \lambda _3^*(\varOmega _0, \gamma _1, \gamma _2)$, then $\sigma _1(\varOmega _0, \lambda , \gamma _1, \gamma _2, 0.5, 0.5, 0) < 0$.

• (b) $\lambda _3^*(\varOmega _0, 0, \gamma _2) = 0$ and $\lambda _3^*(\varOmega _0, \gamma _1, \gamma _2) \rightarrow \lambda _1^0(\varOmega _0)$ as $\gamma _1 \rightarrow \infty$ for all $\gamma _2 > 0$.

A proof of Theorem 4 is given in Sect. C.4.

1.3 Results

Interpreting $\lambda$ as being proportional to the intrinsic growth rate r, we can employ Corollary 1 to arrive at a minimum intrinsic growth rate given in (23) with $\gamma = \gamma _1 H(\gamma _2)$, namely

$$\begin{aligned} r^{**}(\varOmega _0, \gamma _1, \gamma _2) = r^{*}(\varOmega _0, \gamma _1 H(\gamma _2)). \end{aligned}$$

(49)

Similarly, interpreting $\lambda$ as being proportional to the patch diffusion rate D, we can employ Corollary 2 to arrive at a maximum patch diffusion rate, namely

$$\begin{aligned} D^{**}(\varOmega _0, \gamma _1, \gamma _2) = {\left\{ \begin{array}{ll} D^*(\varOmega _0, \gamma _1 H(\gamma _2)); &{} \text{ Type } \text{ I } \text{ DD }\\ D^*(\varOmega _0, \gamma _1 H(\gamma _2))); &{} \text{ Type } \text{ II } \text{ DD }\\ \frac{r \ell ^2}{\lambda _3^*(\varOmega _0, \gamma _1, \gamma _2)}; &{} \text{ CD } \text{ or } \text{ Type } \text{ III } \text{ DD }\\ \end{array}\right. } \end{aligned}$$

(50)

Finally, interpreting $\lambda$ as being proportional to the patch size $\ell$, we present a corresponding minimum patch size required for a prediction of persistence for the theoretical population given a fixed patch diffusion rate and patch intrinsic growth rate:

$$\begin{aligned}&\ell ^{**}(\varOmega _0, \gamma _1, \gamma _2) = \sqrt{\frac{D}{r} \lambda _3^*(\varOmega _0, \gamma _1, \gamma _2)}. \end{aligned}$$

(51)

Using (48) and Theorem 4 yields the following result detailing the connection between the patch size and predictions of persistence from the model.

Corollary 4

Let $\gamma _1, \gamma _2 > 0$ be defined as in Table 13 according to the interface scenario assumed. Then, we have the following:

• (a) if $\ell > \ell ^{**}(\varOmega _0, \gamma _1, \gamma _2)$, then (43) has a unique positive equilibrium, v, that is a global attractor for nonnegative nontrivial solutions of (43);

• (b) if $\ell \le \ell ^{**}(\varOmega _0, \gamma _1, \gamma _2)$, then $v \equiv 0$ is a global attractor for nonnegative nontrivial solutions of (43).

1.4 Proof of Theorem 4

Before we present a proof for Theorem 4, we first present and prove a useful lemma, namely

Lemma 2

Let $g(s) = \sqrt{s} \gamma _1 H(\sqrt{s} \gamma _2)$. Then, for all $\gamma _1, \gamma _2 > 0$ we have the following:

• (a) $\lambda _1(\varOmega _0, g(\lambda ))$ is a strictly increasing function of $\lambda$ for all $\lambda > 0$

• (b) $\lambda _1(\varOmega _0, g(\lambda )) \rightarrow 0 \text{ as } \lambda \rightarrow 0^+$ and $\lambda _1(\varOmega _0, g(\lambda )) \rightarrow \lambda _1^0(\varOmega _0) \text{ as } \lambda \rightarrow \infty$

• (c) $\lambda _1(\varOmega _0, g(\lambda ))$ is a concave function of $\lambda$ for all $\lambda > 0$.

Proof of Lemma 2

To prove (a), we first note that $k_1(s)> k_0(s)> 1; s > 0$ [see Yang and Chu (2016)]. It is then easy to see that

$$\begin{aligned}&\frac{d}{d\lambda } \left[ \lambda _1(\varOmega _0, g(\lambda ))\right] = \lambda _1'(\varOmega _0, g(\lambda )) \frac{\gamma _1 \gamma _2 \left( k_1^2\left( \sqrt{\lambda } \gamma _2\right) -k_0^2\left( \sqrt{\lambda } \gamma _2\right) \right) }{2 k_0^2\left( \sqrt{\lambda } \gamma _2\right) } > 0 \end{aligned}$$

since $\lambda _1(\varOmega _0, \beta )$ is strictly increasing in $\beta$ (see Lemma 1). To show (b), we first consider the Taylor series for $k_0$ and $k_1$ both centered at $s = 0$, namely

$$\begin{aligned} k_1(s)= & {} \frac{1}{s} + O(s) \end{aligned}$$

(52)

$$\begin{aligned} k_0(s)= & {} -\varGamma + \ln (2) - \ln (s) + O(s) \end{aligned}$$

(53)

where $\varGamma$ is Euler’s constant. Thus,

$$\begin{aligned} \underset{\lambda \rightarrow 0^+}{\lim } \lambda _1(\varOmega _0, g(\lambda ))= & {} \lambda _1 \left( \varOmega _0, \underset{\lambda \rightarrow 0^+}{\lim } \gamma _1 \sqrt{\lambda } \frac{k_1\left( \sqrt{\lambda }\gamma _2\right) }{k_0\left( \sqrt{\lambda }\gamma _2\right) } \right) \nonumber \\&\lambda _1 \left( \varOmega _0, \underset{\lambda \rightarrow 0^+}{\lim } \gamma _1 \sqrt{\lambda } \frac{\frac{1}{\sqrt{\lambda } \gamma _2} + O\left( \sqrt{\lambda } \gamma _2\right) }{-\varGamma + \ln (2) - \ln \left( \sqrt{\lambda } \gamma _2\right) + O\left( \sqrt{\lambda } \gamma _2\right) } \right) .\nonumber \\ \end{aligned}$$

(54)

Making the change of variables $t = \sqrt{\lambda } \gamma _2$, (54) becomes

$$\begin{aligned}&\underset{\lambda \rightarrow 0^+}{\lim } \lambda _1(\varOmega _0, g(\lambda )) = \lambda _1 \left( \varOmega _0, \underset{t \rightarrow 0^+}{\lim } \frac{\gamma _1}{\gamma _2} \frac{1 + O\left( t^2\right) }{-\varGamma + \ln (2) - \ln \left( t\right) + O\left( t\right) } \right) = 0 \end{aligned}$$

(55)

since by Lemma 1$\lambda _1(\varOmega _0, \beta ) \rightarrow 0$ as $\beta \rightarrow 0^+$. Also, from Yang and Chu (2016), $H(\sqrt{\lambda }\gamma _2) \rightarrow 1$ as $\lambda \rightarrow \infty$ for fixed $\gamma _2 > 0$ and thus $g(\lambda ) \rightarrow \infty$ as $\lambda \rightarrow \infty$. This fact and Lemma 1 give that $\lambda _1(\varOmega _0, g(\lambda )) \rightarrow \lambda _1^0(\varOmega _0)$ as $\lambda \rightarrow \infty$.

Finally, to show (c), we note that since Lemma 1 gives that $\lambda _1(\varOmega _0, \beta )$ is concave in $\beta$ and $\lambda _1(\varOmega _0, \beta )$ is strictly increasing in $\beta$ it suffices to show that g(s) is concave in s, or equivalently that

$$\begin{aligned} g'(\lambda ) = \frac{\gamma _1 \gamma _2}{2} \left( \frac{k_1\left( \sqrt{\lambda }\gamma _2\right) }{k_0\left( \sqrt{\lambda }\gamma _2\right) } - 1\right) \le \frac{\gamma _1 k_1\left( \sqrt{\lambda }\gamma _2\right) }{\sqrt{\lambda }k_0\left( \sqrt{\lambda }\gamma _2\right) } = \frac{g(\lambda )}{\lambda };\ \lambda > 0. \end{aligned}$$

(56)

It is easy to see that (56) will hold as long as

$$\begin{aligned} \frac{1}{\sqrt{\lambda }} \left( \sqrt{\lambda }\gamma _2 - 2\right) \frac{k_1\left( \sqrt{\lambda }\gamma _2\right) }{k_0\left( \sqrt{\lambda }\gamma _2\right) } -\gamma _2 \le 0;\ \lambda > 0. \end{aligned}$$

(57)

Clearly, (57) will hold if $\sqrt{\lambda }\gamma _2 - 2 \le 0$. Thus, assume $\sqrt{\lambda }\gamma _2 - 2 > 0$. Using the fact that $\frac{k_1\left( \sqrt{\lambda }\gamma _2\right) }{k_0\left( \sqrt{\lambda }\gamma _2\right) } < 1 + \frac{1}{2 \sqrt{\lambda } \gamma _2}$ for all $\lambda , \gamma _2 > 0$ [see Yang and Chu (2016)], (57) becomes

$$\begin{aligned} \frac{1}{\sqrt{\lambda }} \left( \sqrt{\lambda }\gamma _2 - 2\right) \frac{k_1\left( \sqrt{\lambda }\gamma _2\right) }{k_0\left( \sqrt{\lambda }\gamma _2\right) } - \gamma _2< & {} \frac{1}{\sqrt{\lambda }} \left( \sqrt{\lambda }\gamma _2 - 2\right) \frac{2\sqrt{\lambda }\gamma _2 + 1}{2\sqrt{\lambda }\gamma _2} - \gamma _2\nonumber \\= & {} \frac{1}{2 \gamma _2 \lambda } \left( 2\lambda \gamma _2^2 - 3 \sqrt{\lambda } \gamma _2 - 2\right) - \gamma _2\nonumber \\= & {} \frac{1}{2 \gamma _2 \lambda } \left( -3\sqrt{\lambda }\gamma _2 - 2\right) < 0. \end{aligned}$$

(58)

Hence, $g(\lambda )$ is a concave function of $\lambda$ for $\lambda > 0$, which implies $\lambda _1(\varOmega _0, g(\lambda ))$ is a concave function of $\lambda$ for all $\lambda > 0$. $\square$

Now we present a proof of Theorem 4.

Proof of Theorem 4

Define $g(s) = \sqrt{s} \gamma _1 H(\sqrt{s} \gamma _2)$. and let $\gamma _1, \gamma _2 > 0$. From Lemma 2, we have that $\lambda _1(\varOmega _0, g(0)) = 0$, $\lambda _1(\varOmega _0, g(\lambda )))$ is a strictly increasing and concave function of $\lambda$ for $\lambda > 0$, and $\lambda _1(\varOmega _0, g(\lambda )) \rightarrow \lambda _1^0(\varOmega _0)$ as $\lambda \rightarrow \infty$. Also, from the proof of Lemma 2 and the fact that we can choose $\phi > 0;\ x \in \overline{\varOmega }_0$, we have that $\lambda _1'(\varOmega _0, g(0)) > 0$. It is now clear that $\lambda _1(\varOmega _0, g(\lambda )) - \lambda > 0$ for $\lambda > 0$ and small enough. But, since $\lambda _1(\varOmega _0, g(\lambda ))$ is concave, strictly increasing, and bounded in $\lambda$ there exits a unique $\lambda _3^*(\varOmega _0, \gamma _1, \gamma _2) > 0$ such that $\sigma _1(\varOmega _0, \lambda , \gamma _1, \gamma _2, 0.5, 0.5, 0) = \lambda _1(\varOmega _0, g(\lambda )) - \lambda \ge 0$ for $\lambda \in \left[ 0, \lambda _3^*(\varOmega _0, \gamma _1, \gamma _2)\right]$ and $\sigma _1(\varOmega _0, \lambda , \gamma _1, \gamma _2, 0.5, 0.5, 0) = \lambda _1(\varOmega _0, g(\lambda )) - \lambda < 0$ for $\lambda > \lambda _3^*(\varOmega _0, \gamma _1, \gamma _2)$, proving part (a) (see Fig. 3). Part (b) is clear from Lemma 2 and the previous argument. $\square$

Fig. 3

Illustration of Theorem (Color figure online) 4