Qualitative dynamics of planar and spatial Lotka-Volterra and Kolmogorov systems

Ordinary differential equations are an important tool for the study of many real problems. In this thesis we focus in the qualitative dynamics of some ordinary differential systems, particularly, the Lotka-Volterra and Kolmogorov systems. We accomplish the study of some Lotka-Volterra systems on dimension three, which we characterize in two families of planar Kolmogorov systems. We give the complete classification of the global phase portraits in the Poincaré disk for those families. We also analyze the limit cycles of the three-dimensional Kolmogorov systems of degree three which appear through a zero-Hopf bifurcation. Some particular systems that model real problems in the field of population dynamics are also studied

Dynamics of a two prey and one predator system with indirect effect

Altres ajuts: Consellería de Educación, Universidade e Formación Profesional (Xunta de Galicia), grant ED431C 2019/10We study a population model with two preys and one predator, considering a Holling type II functional response for the interaction between first prey and predator and taking into account indirect effect of predation. We perform the stability analysis of equilibria and study the possibility of Hopf bifurcation. We also include a detailed discussion on the problem of persistence. Several numerical simulations are provided in order to illustrate the theoretical results of the paper

Global phase portraits of a predator-prey system

We classify the global dynamics of a family of Kolmogorov systems depending on three parameters which has ecological meaning as it modelizes a predator–prey system. We obtain all their topologically distinct global phase portraits in the positive quadrant of the Poincaré disc, so we provide all the possible distinct dynamics of these systems

Phase portraits of a family of Kolmogorov systems depending on six parameters

Altres ajuts: Consellería de Educación, Universidade e Formación Profesional (Xunta de Galicia), grant ED431C 2019/10We consider a general 3-dimensional Lotka-Volterra system with a rational first integral of degree two of the form H = xi yj zk. The restriction of this Lotka-Volterra system to each surface H(x, y, z) = h varying h ∈ R provide Kolmogorov systems. With the additional assumption that they have a Darboux invariant of the form xl ym est they reduce to the Kolmogorov systems x˙ = x (a0 − µ(c1x + c2z2 + c3z)), z˙ = z (c0 + c1x + c2z2 + c3z)). We classify the phase portraits in the Poincaré disc of all these Kolmogorov systems which depend on six parameters

Global phase portraits of a predator–prey system

Altres ajuts: Consellería de Educación, Universidade e Formación Profesional (Xunta de Galicia) grant ED431C 2019/10We classify the global dynamics of a family of Kolmogorov systems depend-ing on three parameters which has ecological meaning as it modelizes a predator-prey system. We obtain all their topologically distinct global phase portraits in the positive quadrant of the Poincaré disc, so we provide all the possible distinct dynamics of these systems

Predator–Prey Models: A Review of Some Recent Advances

In recent years, predator–prey systems have increased their applications and have given rise to systems which represent more accurately different biological issues that appear in the context of interacting species. Our aim in this paper is to give a state-of-the-art review of recent predator–prey models which include some interesting characteristics such as Allee effect, fear effect, cannibalism, and immigration. We compare the qualitative results obtained for each of them, particularly regarding the equilibria, local and global stability, and the existence of limit cycles

Planar Kolmogorov systems with infinitely many singular points at infinity

Altres ajuts: Consellería de Educación, Universidade e Formación Profesional (Xunta de Galicia), grant ED431C 2019/10We classify the global dynamics of the five-parameter family of planar Kolmogorov systems y˙ = y (b0 + b1yz + b2y + b3z), z˙ = z (c0 + b1yz + b2y + b3z), which is obtained from the Lotka-Volterra systems of dimension three. These systems have infinitely many singular points at infinity. We give the topological classification of their phase portraits in the Poincaré disc, so we can describe the dynamics of these systems near infinity. We prove that these systems have 13 topologically distinct global phase portraits

Study of the Nonelementary Singular Points and the Dynamics Near the Infinity in Predator-Prey Systems

In this chapter, we present the results obtained in two predator-prey systems, paying special attention to the dynamics near the infinity and the nonelementary singular points. First, the desingularization technique known as blow-up technique allows one to study any type of singularities of analytic systems in dimension two even if they are not elementary. In the other hand, the introduction of the Poincaré compactification allows one to accomplish a complete study of the global dynamics of these systems. In addition to the proofs of the results obtained in those cases, we include a survey on the used techniques from a theoretical point of view

Phase portraits of a family of Kolmogorov systems with infinitely many singular points at infinity

Altres ajuts: Consellería de Educación, Universidade e Formación Profesional (Xunta de Galicia) grant ED431C 2019/10We give the topological classification of the global phase portraits in the Poincaré disc of the Kolmogorov systems ẋ=xa+cx+cz+cz, ż=zc+cx+cz+cz, which depend on five parameters and have infinitely many singular points at the infinity. We prove that these systems have 22 topologically distinct phase portraits