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Nonlocal generalized models of predator-prey systems. (English) Zbl 1271.34054
Consider a predator-prey model of the form \[ \begin{aligned} X'&=S(X)-G(X,Y), \\ Y'&=G(X,Y)-M(Y),\end{aligned} \] where \(S,G,M\) are smooth functions. The method of generalized modeling is as follows. Assume that the system has at least one positive equilibrium \((X^*,Y^*)\). Using the transformation \(x=X/X^*\), \(y=Y/Y^*\), the equilibrium is moved to \((x^*,y^*)=(1,1)\) and system (1) can be rewritten as \[ \begin{aligned} x'&=\beta_1[s(x)-g(x,y)], \\ y'&=\beta_2[g(x,y)-m(y)] \end{aligned} \] with appropriately defined constants \(\beta_1,\beta_2\) and functions \(s,g,m\). Here, \(\beta_1,\beta_2\) are the so-called scale parameters. The Jacobian of the system at \((1,1)\) is associated with the so-called elasticities defined by \[ s_x=\partial_x(s(x))|_{x=1},\;g_x=\partial_x(g(x,y))|_{(x,y)=(1,1)}, \]
\[ g_y=\partial_y(g(x,y))|_{(x,y)=(1,1)},\;m_y=\partial_y(m(y))|_{y=1}. \] The scale parameters and elasticities are termed generalized parameters. Stability analysis, bifurcation analysis, and other methods can be applied to the model in the generalized parameter space and obtain results that are valid for a wide class of models.
In this paper, the method of generalized modeling is extended to systems with periodic solutions. Let \(\gamma(t)=(\gamma_1(t),\gamma_2(t))\) be a positive periodic orbit of system (1) with period \(T\). Similarly, using the transformation \(x=X/\gamma_1\), \(y=Y/\gamma_2\), the periodic orbit is reduced to the point \((1,1)\). System (1) now takes the form \[ \begin{aligned} x'&=\beta_s[s(x)-x]-\beta_1[g(x,y)-x], \\ y'&=\beta_2[g(x,y)-y]-\beta_m[m(y)-y]. \end{aligned} \] Then different methods, Floquet multipliers, Fourier series and statistical sampling, are used to obtain results on the stability of periodic solutions.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
92D25 Population dynamics (general)
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