Lotka–Volterra Competition Model with Cross-Diffusion

Abstract

Consider the Lotka–Volterra competition model

$$\begin{cases}\frac{du}{dt}=cu-\gamma_{11}u^2-\gamma_{12}uv&\text{for}\ 0 < t <\infty, \\\noalign{\vspace{3pt}}\frac{dv}{dt}=dv-\gamma_{21}uv-\gamma_{22}v^2&\text{for}\ 0 < t <\infty,\end{cases}$$

(15.1)

for two species, say A and B, where u=u(t) denotes the total population of A, and v=v(t) the total population of B, respectively, at time t. Let the conditions

$$d{\gamma_{12}}>c{\gamma_{22}}\quad\text{and}\quad c{\gamma_{21}}>d{\gamma_{11}}$$

(15.2)

be satisfied. We note that these conditions imply γ ₁₁ γ ₂₂<γ ₁₂ γ ₂₁, that is, interspecies competition is stronger than intraspecies competition. Then the equations

$$\begin{cases}u(c-\gamma_{11}u-\gamma_{12}v)=0,\\v(d-\gamma_{21}u-\gamma_{22}v)=0\end{cases}$$

have four nonnegative solutions

$$O=(0,0),\qquad \overline{U}_u=(c/\gamma_{11},0),\qquad \overline{U}_v=(0,d/\gamma_{22}),\qquad \overline{U}=(\overline{u},\overline{v}),$$

(15.3)

where $\overline{u}=(c\gamma_{22}-d\gamma_{12})/D$ and $\overline{v}=(d\gamma_{11}-c\gamma_{21})/D$ with D=γ ₁₁ γ ₂₂− γ ₁₂ γ ₂₁<0 (due to (15.2)). The solutions $O,\,\overline{U}_{u}$ , and $\overline{U}_{v}$ are extinct solutions; meanwhile, $\overline{U}$  is a coexistent solution. By the theory ordinary equations, it is easy under (15.2) to see that the coexistent solution $\overline{U}$ is always unstable. This means that in the case where interspecies competition dominates intraspecies one, it is impossible to explain the coexistence of the species A and B by the ordinary system (15.1).