## 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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### References:

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