| A consumer-resource reaction-diffusion model with a single consumer species was proposed and experimentally studied by Zhang et al.[40].Analytical study on its dynamics was further performed in[16].Firstly,we settle a conjecture in[16].By developing suitable integral inequalities,we use Lyapunov functional method[37]com-bined with the theory of chain recurrent sets for asymptotically autonomous semiflows[27]to prove global stability of the unique non-constant equilibrium solution in hetero-geneous environments for small yield rateγ.We then study a multi-species consumer-resource model where all the consumer species compete with each other through depression of the limited resources by con-sumption and there is no direct competition between them.We show that in this case,all consumer species persist uniformly,which implies that”competition exclusion”phe-nomenon will never happen.This is drastically different from the properties of Lotka Volterra systems.At the same time,we improve the dynamic properties of the multi-species consumer-resource reaction-diffusion model in the homogeneous environment.For the heterogeneous environment,the situation is more complex.In this pa-per,under certain conditions,we study the existence and uniqueness of the coexistence steady state of the two-species consumer-resource reaction-diffusion model.We divide the area to which the yield rate（γ1,γ2）belongs,and divide the coexistence solutions of the system into three categories.We construct a parabolic monotone dynamical system as an auxiliary system.The existence of the coexistence steady state is proved by using the method of upper and lower solutions,and the uniqueness of the coexistence steady state is proved by monotonic dynamic flow theory. |
