Existence, uniqueness and asymptotic behavior of solutions for a nonsmooth producer-grazer system with stoichiometric constraints

Abstract

A reaction-diffusion-ODE system of stoichiometric producer-grazer type is considered in this paper. Since the system has nonsmooth nonlinearity, it is shown that the system has new dynamics different from the smooth case. We construct various types of discontinuous steady states and investigate their asymptotic behavior. In addition, the steady-state bifurcation near a constant solution is studied by treating the diffusion coefficient as a bifurcation parameter and the existence of Hopf bifurcation is derived. Our results cover the case where the number of positive equilibria of the kinetic system (i.e. without diffusion) changes from one to three in the spatial interval. Finally, some numerical simulations are given to illustrate the theoretical results.

Keywords:

• Stoichiometric producer-grazer system
• stationary solution with jump discontinuities
• asymptotic behavior
• bifurcation

2020 MSC:

• 35B36
• 35B40
• 35M33
• 35B32
• 92D25

Acknowledgments

The authors would like to give their sincere thanks to Professor Meirong Zhang and the anonymous referee for their valuable suggestions leading to the improvement of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data can be made available on reasonable request.

Additional information

Funding

This study was funded by Shuimu Tsinghua Scholar Program [grant number 2021SM062], National Natural Science Foundation of China [grant numbers 11790273, 11901446] and Postdoctoral Science Foundation of China [grant numbers 2022M711854, 2021T140530].