Long-term coexistence for a competitive system of spatially varying gradient reaction–diffusion equations

Abstract

Spatial distribution of interacting chemical or biological species is usually described by a system of reaction–diffusion equations. In this work we consider a system of two reaction–diffusion equations with spatially varying diffusion coefficients which are different for different species and with forcing terms which are the gradient of a spatially varying potential. Such a system describes two competing biological species. We are interested in the possibility of long-term coexistence of the species in a bounded domain. Such long-term coexistence may be associated either with a periodic in time solution (usually associated with a Hopf bifurcation), or with time-independent solutions. We prove that no periodic solution exists for the system. We also consider some steady states (the time-independent solutions) and examine their stability and bifurcations.

Introduction

A model describing interaction of two competing biological species with a possibility of non-homogeneous spatial distribution usually consists of a coupled set of reaction–diffusion equations with logistic population growth and some form of competition (see e.g. [9], [7]). Based on an idea that “the interaction is (to a degree) unknown in detail”, Norbury and Wake [8] incorporated a quadratic dependence for the interaction logistic terms into a mathematical model. To describe the spatial spread of two competing species–US original grey and UK indigenous red squirrels in the UK–they suggested a system of two reaction–diffusion equations

$$u_t-u_{xx}=\lambda u(1-u^2-a^2{v}^2)\text{,}{v}_t-v_{xx}=\lambda v(b^2-a^2{u}^2-c^2{v}^2)\text{,}$$

where $u(x,t)$ and $v(x,t)$ are densities of the two species, $a,b,c$ are positive constants and $\lambda$ is a positive parameter; of course, only the case $u,v\ge 0$ is feasible for a population or concentration model. However, in some cases both positive and negative solutions can be made physically relevant after scaling of the form $u=\alpha +\beta \tilde{u}$, $v=\gamma +\delta \tilde{v}$ (since $u_x=v_x=0$ translates to ${\tilde{u}}_x={\tilde{v}}_x=0$, and $u_t=\beta {\tilde{u}}_t$, etc.) which transforms the reaction term without changing the rest of the equations. Depending on the nature of an application (or the boundary conditions), we may wish to consider either strictly positive (non-negative) solutions, or solutions of both signs.

A remarkable feature of this model is that the system (1.1) is a gradient system: the right parts of the Eq. (1.1) are partial derivatives, $G_u=\lambda u(1-u^2-a^2{v}^2)$ and $G_v=\lambda v(b^2-a^2{u}^2-c^2{v}^2)$ respectively, of the potential

$$G(u,v)=\frac{\lambda }{2}(u^2-\frac{1}{2}{u}^4-a^2{u}^2{v}^2+b^2{v}^2-\frac{1}{2}{c}^2{v}^4)\text{.}$$

Further, there is a symmetry, or a group action $u\to -u$ and $v\to -v$ under which the potential (and hence the system) is invariant. The model may be generalised to the multi-dimensional case in a straightforward way.

In modelling situations gradient reaction–diffusion systems can and do arise from physically or physiologically derived situations when there is some type of “overall global conservation” occurring. In population biology, from which this model arose, this can occur when species are competing with each other for resources. The model in Eq. (1.1) was determined both by looking at the data and by consideration of the specific growth rate of each component species in the presence of the other [9]. Subsequent scaling of the variables enabled it to be cast in the form of a gradient system. Care must be taken in doing this to ensure that the boundary conditions are not inadvertently altered so that the system no longer satisfies the conditions of the results in Section 2. In the specific case of the competing squirrel populations this did not occur as it involved just a simple scaling of the populations. That is, it is a “robust gradient system”.

Norbury and Wake [8] focused on the possibility of long-term coexistence of two competing species, that is possibility of static (time-independent) or periodic solutions. A time-independent spatial distribution of the densities of the two species $u\left(x\right)$ and $v\left(x\right)$ satisfies the equations

$$\frac{d^{2}u}{d{x}^2}=-\lambda u(1-u^2-a^2{v}^2)\text{,}\frac{d^{2}v}{d{x}^2}=-\lambda v(b^2-a^2{u}^2-c^2{v}^2)\text{,}$$

defined on an open interval $0<x<1$ (the actual length of the domain was scaled into $\lambda$). To close the system (1.2), boundary conditions should be added. The no-flux boundary conditions

$$u_x\left(0\right)=v_x\left(0\right)=0\text{,}{u}_x\left(1\right)=v_x\left(1\right)=0$$

reflect a situation of impermeable boundaries. Under a “hostile environment” assumption the Dirichlet boundary conditions

$$u\left(0\right)=v\left(0\right)=0\text{,}u\left(1\right)=v\left(1\right)=0$$

can be used. The latter case was considered by Norbury and Wake [8].

The system (1.2) is a Hamiltonian system with independent variable $x$: the Eq. (1.2) are the Euler–Lagrange equations for the functional

$$I={\int }_0^1(\frac{1}{2}(u_x^2+v_x^2)-G(u,v))dx\text{.}$$

The function

$$E\left(x\right)=\frac{1}{2}(u_x^2+v_x^2)+G(u,v)$$

is the first integral of the system (1.2): that is for any solution of the system (1.2) the function $E\left(x\right)$ is constant for all $x$. In particular in the case of one spatial dimension for the homogeneous Neumann boundary conditions $E\left(x\right)=G(u\left(0\right),v\left(0\right))=G(u\left(1\right),v\left(1\right))$ for all $x\in [0,1]$.

In this work we consider the existence of long-term coexisting solutions–both periodic in time and steady-state coexistence–of two species in a multi-dimensional bounded simply connected region $\Omega \subset R^n$. We also extend Eq. (1.1) so that different species diffusions and more general interaction potential functions $W(x,u,v)$ are allowed. In Section 2 we consider the possibility of periodicity in time for this system, and show that no periodicity is possible. We then go on in Sections 3 Steady-state solutions, 4 Spatially uniform steady states to consider steady states and to classify them as regards to stability. In Sections 5 Explicit solutions, 6 The special case of a homogeneous potential, 7 Bifurcations of non-coexisting solutions we consider some explicit solution which may exist for biological system with competition, and their bifurcations.