Bifurcation analysis of a generalized Gause model with prey harvesting and a generalized Holling response function of type III. (English) Zbl 1217.34080

This paper studies a generalized Gause model with prey harvesting and a generalized Holling response function of type III:
\[ \dot{x} = \rho x(1-x) - y p(x) - \lambda,\quad \dot{y} = y (-\delta + p(x)), \tag{1} \]
where \(x\geq 0\), \(y\geq 0\), and
\[ p(x) = {x^2 \over \alpha x^2 + \beta x + 1}. \tag{2} \]
This basic result is a bifurcation diagram to equation (1).
The authors show that the \(x\)-axis of system (1) is invariant. The system has 2 singular points, \(C\) and \(D\), on the positive \(x\)-axis for \(\rho > 4 \lambda\) and no equilibrium for \(\rho < 4 \lambda\). The two points merge in a saddle-node for \(\rho = 4 \lambda\). In the first quadrant, there is at most one singular point \(E\) which is always of anti-saddle type (i.e., a node, focus, weak focus or center). The singular point \(E\) disappears from the first quadrant by a saddle-node bifurcation by merging, with either \(C\), or \(D\). The point \(E\) can undergo a Hopf bifurcation of order at most two. When the order is two, the second Lyapunov coefficient is positive (the weak focus is repelling).


34C60 Qualitative investigation and simulation of ordinary differential equation models
34C23 Bifurcation theory for ordinary differential equations
92D25 Population dynamics (general)
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