Higher codimension bifurcation analysis of predator-prey systems with nonmonotonic functional responses. (English) Zbl 1453.34074
Summary: In this paper, we study the dynamic behaviors of a predator-prey system with a general form of nonmonotonic functional response. Through analysis, it is found that the system exists in extinction equilibrium, boundary equilibrium and two positive equilibria, one or no positive equilibrium. Furthermore, the conditions are given such that the boundary equilibrium is a saddle, node or a saddle-node point of codimension 1, 2 or 3. Then, some conditions are obtained so that the unique positive equilibrium of the system is a cusp point of codimension 2, 3 and higher or a saddle-node one of codimension 1 or 3, or a nilpotent saddle-node of codimension 4. When there are two positive equilibria in the system, the equilibrium with a large number of preys is a saddle point. For the other one, the system may undergo Hopf bifurcation. To verify our conclusion, we consider the functional response function in the literature [H. Zhu et al., SIAM J. Appl. Math. 63, No. 2, 636–682 (2002; Zbl 1036.34049); D. Xiao and S. Ruan, Int. J. Bifurcation Chaos Appl. Sci. Eng. 11, No. 8, 2123–2131 (2001; Zbl 1091.92504)]. Finally, we give a brief discussion and numerical simulation.
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C20 | Transformation and reduction of ordinary differential equations and systems, normal forms |
| 34D20 | Stability of solutions to ordinary differential equations |
| 92D25 | Population dynamics (general) |
Keywords:
predator-prey model; nonmonotonic functional response function; bifurcation; normal form; higher codimensionReferences:
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