We consider a space dependent predator-prey system with diffusing predator and sedentary prey obeying a local dynamics which is specific but not strictly of a Lotka-Volterra type. In our model, we assume the Malthusian parameter in the relative rate of the increment in the resource to be space dependent. This coefficient represents the environmental variability of the ecologically active medium and it introduces a source of spatial inhomogeneity which is characteristic of natural systems. Also the diffusion coefficient of the predator is considered space dependent, except in the analysis of global stability.
Systems of this type have been widely investigated in the literature, the majority of authors assuming diffusing predators and preys (or both competitors) and a Lotka-Volterra local dynamics. Some authors consider the degenerate case that only one component diffuses and the other one is sedentary. In these models the local dynamics are of a Lotka-Volterra type and the coefficients may be space dependent. The main difference between our model and the previous ones is the assumption of a more general local dynamics.
In Sect. 2, the basic ecological assumptions and the evolution equations are presented. In Sect. 3, we perform an existence and stability analysis of the steady state solutions as a function of the rate of increment in resource. Our analysis relies on sub- and supersolution techniques, on finding criteria for existence in terms of the fundamental eigenvalue of elliptic equations, and on the construction of a Lyapunov function. In Sect. 4, a case study relative to an algae-polychaete community is worked out, which permits us to compare the results of the mathematical model with an experimentally observed situation.