Abstract
Abstract In this manuscript, we first consider the diffusive competition and cooperation system subject to Neumann boundary conditions without delay terms and get the conclusion that the unique positive constant equilibrium is locally asymptotically stable. Then, we study the diffusive delayed competition and cooperation system subject to Neumann boundary conditions, and the existence of Hopf bifurcation at the positive equilibrium is obtained by regarding delay term as the parameter. By the theory of center manifold and normal form, an algorithm for determining the direction and stability of Hopf bifurcation is derived. Finally, some numerical simulations and summarizations are carried out for illustrating the theoretical analytic results.
Highlights
In the past few decades, delay differential equations that change in time and involve delays have become a hot research topic
Much commonness is reflected between species that co-evolve in nature and different enterprises that co-exist in economic society, so numerous researchers have widely presented the competition and cooperation model of the enterprises [17,18], which are governed by the following ordinary differential equation: ( )
This paper aims to investigate the stability of equilibria and the properties of Hopf bifurcation at the unique positive constant equilibrium of system (1.4)
Summary
In the past few decades, delay differential equations that change in time and involve delays have become a hot research topic. Much commonness is reflected between species that co-evolve in nature and different enterprises that co-exist in economic society, so numerous researchers have widely presented the competition and cooperation model of the enterprises [17,18], which are governed by the following ordinary differential equation:. Considering the heterogeneity of enterprise spatial distribution, motivated by the present situation stated above, we take the inhomogeneity of the spatial distribution into account and obtain the following competition and cooperation system incorporating diffusion and delay subject to Neumann boundary conditions.
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