A global mathematical investigation of a predator-prey model

We construct a global bifurcation diagram of the plane differential system ẋ = x(1− x)− xy/(a+ x), ẏ = y(δ − βy/x), x(t) > 0, y(t) > 0, a > 0, δ > 0, β > 0, which describes the predator-prey interaction. In [1] was considered the following planar differential systems which describes a mathematical predator-prey model ẋ = rx(1 − x K )− mx b+x y, ẏ = sy(1− y hx ). (1) Here x(t) > 0, y(t) > 0 — the population densities of the prey and the predator respectively and r, s,K, h,m — positive parameters ( for more details see [1]). After the variables and parameters scaling the system takes the form ẋ = x(1 − x)− xy a+x , ẏ = y(δ − β y x ), x(t) > 0, y(t) > 0, a > 0, δ > 0, β > 0. (2) System (2) was investigated with the help of the qualitative theory of the ordinary differential equations in [1] where it was demonstrated some properties of the system. For example, it was shown that there is an positively invariant region in the phase plane and it was found how many steady states are located there and it were found their types and so on. During the investigation the authors of [1] studied some bifurcations of codim 1 and 2: the turning point bifurcation, the cusp bifurcation, the Andronov-Hopf bifurcation, the BogdanovTakens bifurcation, the loop bifurcation. A special attention was given to the study of periodic solutions. The result of [1] were extended in [2]. The work is supported by RFBR (09-01-00070) and by Siberian branch of RAS (interdisciplinaire integration projects no. 107, 119 The work is supported by RFBR (09-01-00070) and by Siberian branch of RAS (interdisciplinaire integration projects no. 107, 119