Qualitative and numerical analysis of a class of prey-predator models
A problem of the dynamics of prey-predator populations raised by D'Ancona in a letter to Volterra is considered: the competition between the predators of the same as well as of different species. Two cases are investigated: In the first one the predator is ''blind'', while in the second one its behaviour is ''intelligent''. The corresponding models are of the form: \[ \dot x_ 1=x_ 1(k_ 1-a_{12}-a_{13}x_ 3-ax_ 1), \] \[ \dot x_ 2=x_ 2(-k_ 2+a_{12}x_ 1-f(x_ 1)x_ 3-g_ 1(x_ 1)x_ 2), \] \[ \dot x_ 3=x_ 3(-k_ 3+a_{13}x_ 1-f(x_ 1)x_ 2-g_ 2(x_ 1)x_ 3). \] Intelligent behaviour of the predator is represented by such an f for which \(f(x_ 1)>0\) \((x_ 1\geq 0)\) and f is nonincreasing. Mechanical or blind behaviour of the predator is modelled by \(f(x_ 1):=hx_ 1\), \(g_ 1(x_ 1):=bx_ 1\), \(g_ 2(x_ 1):=cx_ 1\) \((h,b,c>0).\) The investigated properties are existence and stability of equilibria, existence of limit cycles and coexistence. Both, qualitative and numerical methods are used. As a preparation, one prey - one predator models with blind and with intelligent behaviour of the predator are analyzed. Next, the full three-dimensional model with blind behaviour and one of its reduced forms are investigated, and then the three-dimensional model with intelligent behaviour. The main conclusion is that although the answer to the original biological problem highly depends on the values of the parameters (at least, within the present framework of the present models) the paper provides a suitable starting point for further investigations.
scientific article; zbMATH DE number 3968415