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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).

MSC:

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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