Global dynamics of a diffusive competition model with habitat degradation

Abstract

In this paper, we propose a diffusive competition model with habitat degradation and homogeneous Neumann boundary conditions in a bounded domain that is partitioned into the healthy region (undisturbed habitat) and the degraded region (due to anthropogenic habitat disturbance). Species follow the Lotka-Volterra competition in the healthy region while in the degraded region species experience only exponential decay (not necessarily at the same rate). This setup is novel in that it requires no positivity assumption on the environmental heterogeneity, either absolute or on average, which would be far too restrictive for the study of the effects of habitat degradation. We rigorously show competitive exclusion and coexistence via global stability analysis. A remarkable finding is that the quality heterogeneity of landscapes can lead to the competitive exclusion of the slower species by the faster species. This result is robust as long as the degraded region has positive area, and moreover is at odds with classical results predicting the deterministic extinction of the stronger species. On the other hand, if the degraded region has intermediate negative effect on the faster competitor, species can coexist. Differing from comparable existing results, coexistence does not rely on a limit as the diffusion coefficients tend to zero or infinity. Together, these results imply that coexistence is always a possibility under this basic, yet general, configuration, providing insights into the varying impacts found through empirical study of habitat loss and fragmentation on species.

Appendix

In this section, we collect results related to some fundamental eigenvalue problems.

1.1 An auxiliary eigenvalue problem

Let $m\in L^{\infty }(\Omega )$ and consider the problem

$$\begin{aligned} { {\left\{ \begin{array}{ll} \Delta \phi + \lambda m \phi = 0 &{} \quad \text {in}\quad \Omega , \\ \frac{\partial \phi }{\partial {\mathbf {n}}} = 0 &{} \quad \text {on}\quad \partial \Omega . \end{array}\right. } }\end{aligned}$$

(A.1)

If there exists a value $\lambda _1 (m)$ and a positive function $\phi _1$ solving (A.1), we call $\lambda _1(m)$ the principal eigenvalue to problem (A.1). The following is a well known result and a good discussion of this problem can be found in (Cantrell and Cosner 2003). The main result is the following, with the statement taken from (Ni 2001, Chapter 4).

Proposition A.1

Let $m\in L^{\infty }(\Omega )$. Problem (A.1) has a nonzero principal eigenvalue $\lambda _1 (m)$ if and only if m changes sign and $\int _\Omega m \ne 0$. More precisely,

• (i) $\int _\Omega m < 0 \Rightarrow \lambda _1 (m) > 0$ ;

• (ii) $\int _\Omega m > 0 \Rightarrow \lambda _1 (m) < 0$;

• (iii) $\int _\Omega m = 0 \Rightarrow 0$ is the only principal eigenvalue.

One can see how this relates to the statement of Proposition A.2: when the average heterogeneity is positive, $\lambda _1 < 0$ and we always have a positive eigenvalue $\mu _1$ to problem A.2. On the other hand, when the average heterogeneity is negative, $\lambda _1 > 0$ and the sign of the eigenvalue $\mu _1$ to problem A.2 depends on the relationship between the size of diffusion d and the size of $\lambda _1$.

1.2 A related eigenvalue problem

Let $m\in L^{\infty }(\Omega )$ and consider the following eigenvalue problem:

$$\begin{aligned} { {\left\{ \begin{array}{ll} d \Delta \phi + m \phi + \mu \phi = 0 &{} \quad \text {in}\quad \Omega , \\ \frac{\partial \phi }{\partial {\mathbf {n}}} = 0 &{} \quad \text {on}\quad \partial \Omega . \end{array}\right. } }\end{aligned}$$

(A.2)

It seems self evident that this problem is closely related to problem A.1. We call $\mu _1(d,m)$ a principal eigenvalue for problem (A.2) whenever there exists a solution $\phi _1 \in C^{+} ( {\overline{\Omega }})\setminus \{0\}$.

It is well-known that this problem has a unique principal eigenvalue admitting the usual variational characterization:

$$\begin{aligned} { \mu _1 (d,m) = \inf \left\{ \int _\Omega \left[ d \left| \nabla \phi \right| ^{2} - m \phi ^2 \right] dx : \phi \in H^{1} (\Omega ), \int _\Omega \phi ^2 dx = 1 \right\} . } \end{aligned}$$

(A.3)

The following proposition highlights some of the classical properties of this eigenvalue. Recall $\lambda _{1}(m)$ from Subsect. A.1.

Proposition A.2

Suppose $m\in L^{\infty }(\Omega )$ is not a constant function. Then the following hold.

• (i) $\int _\Omega m \ge 0 \ \Rightarrow \ \mu _1 (d,m) < 0 \ \text { for all } d > 0$.

• (ii) $\int _\Omega m < 0 \ \Rightarrow \$ ${\left\{ \begin{array}{ll} \mu _1 (d,m)< 0, \quad \text {if} \quad d < \lambda _1 ^{-1} (m), \\ \mu _1 (d,m) = 0, \quad \text {if} \quad d = \lambda _1^{-1} (m) , \\ \mu _1 (d,m)> 0, \quad \text {if} \quad d > \lambda _1 ^{-1} (m). \end{array}\right. }$

• (iii) $\mu _1 (d, m)$ is strictly increasing and concave with respect to $d>0$.

• (iv) $\mu _1 (d , m ) < \mu _1 (d, {\tilde{m}})$ whenever $m \gneqq {\tilde{m}}$.

• (v) $\mu _1 (d,m)$ is continuous in m with respect to $L^\infty (\Omega )$.

Points (i)–(iv) appear in (Ni 2001), however point (v) is not explicitly discussed. Point (v) is proven in (Hess 1991) when $m_n \rightarrow m$ in $C ( {\overline{\Omega }} )$, whereas a weakened regularity case (with respect to $L^p (\Omega ),\ p > N/2$) is discussed in detail in (Daners 1997; Fleckinger and Lapidus 1986; Hess 1985), for example.