Smoothly intertwined basins of attraction in a prey-predator model
The relative alignment of basins of attraction in a prey-predator model which is of interest for the biological control of parasites in agriculture is considered. The model is assumed to be given by the two- dimensional system of Lotka-Volterra type \[ \dot x= (h_1- ky)x, \quad \dot y= (-h_2+ kx- f(x) y)y, \qquad (x,y\geq 0), \tag{1} \] where \(k, h_1, h_2>0\), for \(x\geq 0\), \(f(x)\) behaves likewise \(bx^3/ (x^5 +c)\). By suitable choices of parameters, one can achieve the system (1) to have three equilibria in \(\{ x>0\), \(y>0\}\): two sinks \(L\), \(R\) and a saddle \(S\). It is shown that if the system (1) has no periodic orbits then the common boundary of the basins of attraction of \(L\) and \(R\) generically intersects the line \(\{y= h_1/ k\}\) and the curve \(\{y= (kx- h_2)/ f(x)\}\) in countably many points, accumulating at \((0, h_1/ k)\) and \((h_2/ k, 0)\), respectively. Such behavior of the boundary leads to the intertwining of the basins of attraction of \(L\), \(R\) and makes difficult the prediction of which final equilibrium (\(L\) or \(R\)) will be attained by the system. Refining of the model by introducing a friction between preys is discussed.
scientific article; zbMATH DE number 790355