On the influence of diffusion on the competition between bacteria and innate immune system of invertebrates (Q619587)

To study the influence of diffusion on the competition between bacteria and the innate immune system of invertebrate animals, the author considers a 2 by 2 system of reaction-diffusion equations, \[ \begin{cases} {\partial B\over \partial t}=aB+\gamma_1\Delta B-\left({a\over k}B+{eI-\gamma\over k_1+B}\right)B \\ {\partial I\over \partial t}=s-cI+dB-\gamma_2\Delta I \end{cases} \] for \((x,t)\in \Omega\times \mathbb{R}^+\), with initial conditions \[ B(x,0)=B_0(x)>0, \;I(x,0)=I_0(x)>0 \] for \(x\in \Omega\), and boundary conditions \[ \beta^*B+(1-\beta^*)\nabla B\cdot {\mathbf n}=\beta^*B^*, \;\beta^*I+(1-\beta^*)\nabla I\cdot {\mathbf n}=\beta^*I^* \] on \(\partial \Omega\). Here \(B(x,t)\) and \(I(x,t)\) denote the concentration of the bacteria and that of the innate immune cells respectively. All parameters involved in the system are positive constants. The existence of absorbing sets is provided and the stability of nonnegative equilibrium solutions is analyzed. In addition, the author also studies the stability switching properties due to diffusion. More specifically, the change of diffusion parameters can drive the nonnegative equilibria from being stable to unstable and vice versa.