Bogdanov-Takens Bifurcation in a Leslie Type Tritrophic Model with General Functional Responses
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Bogdanov-Takens Bifurcation in a Leslie Type Tritrophic Model with General Functional Responses Gamaliel Blé1 · Miguel Angel Dela-Rosa2
Received: 25 June 2019 / Accepted: 14 November 2019 © Springer Nature B.V. 2019
Abstract The dynamics of a differential system modeling a tritrophic food chain of Leslie type is analyzed. It is assumed that the prey has general growth rate and the predator (superpredator) population eats the prey (predator) through a general functional response. The parameter conditions that ensure a coexistence equilibrium point, where the differential system exhibits a codimension 2 Bogdanov-Takens bifurcation (BTb), is given. Some numerical examples where the functional responses are Holling type and the prey population has logistic growth rate are shown. The techniques that have been used to obtain these results may be applied to ecological models with other functional responses types. Keywords Bogdanov-Takens-bifurcation · Leslie type tritrophic model · Holling Functional responses Mathematics Subject Classification (2010) 37G15 · 37C75 · 92D25
1 Introduction One of the predator-prey differential systems that had provided a well adapted model in the sense that several dynamics with either ecological or biological sense can be studied are those of Leslie type. Such differential systems are characterized by the following: the prey population with density x, has logistic growth rate with carrying capacity R and growth rate ρ in the absence of predator population. Also in this system one has a generalist predator population with density y, with grow rate s and that eats the prey by means of a functional
B M.A. Dela-Rosa
[email protected] G. Blé [email protected]
1
División Académica de Ciencias Básicas, UJAT, Km 1, Carretera Cunduacán–Jalpa de Méndez, Cunduacán, Tabasco, c.p. 86690, Mexico
2
División Académica de Ciencias Básicas, CONACyT-UJAT, Km 1, Carretera Cunduacán–Jalpa de Méndez, Cunduacán, Tabasco, c.p. 86690, Mexico
G. Blé, M.A. Dela-Rosa
response f (x), and it has a carrying capacity that is proportional to the population density of prey with proportionality constant α that measure the food quality of the prey for conversion into predator births (see [4], [11] and the references therein). Specifically, these systems have the form x x˙ = ρx 1 − − yf (x), R (1) y y˙ = y s 1 − . αx On the other hand, with the aim to attempt the analysis of global stability for predatorprey systems, one may ask whether it is possible to guarantee the existence of an equilibrium point at which these systems undergo a codimension two (nondegenerate) Bogdanov-Takens bifurcation (for short, this nondegenerate bifurcation will be denoted by BTb) from which one has a rich dynamics due to the presence of several bifurcation types: a saddle-node bifurcation, a Hopf bifurcation and a saddle homoclinic bifurcation, [10, 15]. The parameter center manifold theory and the normal form method have been used to guarantee the existence of a Bogdanov-Takens bifurcation at these systems. We cite some wo
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