Analysis of some dynamical systems inspired by ecological interactions

A community is a collection of populations of different species living in the same geographical area. Species interact with each other in the community and this interaction affects species distribution, abundance, and even evolution. Species interact in various ways, for instance through competition, predation, parasitism, mutualism, and commensalism. We have two focuses in this thesis. One focus is analyzing the dynamical behaviors of the discretization systems of the Lotka-Volterra predator-prey model. It is well known that the dynamics of the logistic map is more complex compared with logistic differential equation. Period doubling and the onset of chaos in the sense of Li-York occur for some values. Inspired by this, we analyze the dynamical behaviors of the discretization systems of the Lotka-Volterra predator-prey model (articles I and II). In article I, we show that the system undergoes fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation, and has a stable invariant cycle in the interior of $R_+^2$ for some parameter values. In article II, we show that the unique positive equilibrium undergoes flip bifurcation and Neimark-Sacker bifurcation. Moreover, system displays much interesting dynamical behaviors, including period-5, 6, 9, 10, 14, 18, 20, 25 orbits, invariant cycles, cascade of period-doubling, quasi-period orbits and the chaotic sets. We emphasize that the discretization of continuous models (articles I and II are not acceptable as a derivation of discrete predator-prey models. A discrete predator-prey model is also formulated in Section 2. We analyze the dynamics (articles I and II) from the mathematical point of view instead of biological point of view. The other focus is disease-competition in an ecological system. We propose a model combining disease and competition and study how a disease affects the two competing species (article III). In our model, we assume that only one of the species is susceptible to an SI type disease with mass action incidence, and that infected individuals do not reproduce but suffer from additional disease induced death. We further assume that infection does not reduce the competitive ability of the infected. We show that infection of the superior competitor enables the inferior competitor to coexist, either as a stable steady state or limit cycle. In the case where two competing species coexist without the disease, the introduction of disease is partially determined by the basic reproduction number. If the reproduction number is less than 1, the disease free coexistence equilibrium is globally asymptotically stable. If the basic reproduction number is larger than 1, our system is uniformly persistent. The unique coexisting endemic disease equilibrium exists and is globally stable under certain conditions. However, infection of the inferior competitor does not change the outcome.A community is a collection of populations of different species living in the same geographical area. Species interact with each other in the community and this interaction affects species distribution, abundance, and even evolution. Species interact in various ways, for instance through competition, predation, parasitism, mutualism, and commensalism. We have two focuses in this thesis. One focus is analyzing the dynamical behaviors of the discretization systems of the Lotka-Volterra predator-prey model. It is well known that the dynamics of the logistic map is more complex compared with logistic differential equation. Period doubling and the onset of chaos in the sense of Li-York occur for some values. Inspired by this, we analyze the dynamical behaviors of the discretization systems of the Lotka-Volterra predator-prey model (articles I and II). In article I, we show that the system undergoes fold bifurcation, flip bifurcation and Neimark-Sacker bifurcation, and has a stable invariant cycle in the interior of $R_+^2$ for some parameter values. In article II, we show that the unique positive equilibrium undergoes flip bifurcation and Neimark-Sacker bifurcation. Moreover, system displays much interesting dynamical behaviors, including period-5, 6, 9, 10, 14, 18, 20, 25 orbits, invariant cycles, cascade of period-doubling, quasi-period orbits and the chaotic sets. We emphasize that the discretization of continuous models (articles I and II are not acceptable as a derivation of discrete predator-prey models. A discrete predator-prey model is also formulated in Section 2. We analyze the dynamics (articles I and II) from the mathematical point of view instead of biological point of view. The other focus is disease-competition in an ecological system. We propose a model combining disease and competition and study how a disease affects the two competing species (article III). In our model, we assume that only one of the species is susceptible to an SI type disease with mass action incidence, and that infected individuals do not reproduce but suffer from additional disease induced death. We further assume that infection does not reduce the competitive ability of the infected. We show that infection of the superior competitor enables the inferior competitor to coexist, either as a stable steady state or limit cycle. In the case where two competing species coexist without the disease, the introduction of disease is partially determined by the basic reproduction number. If the reproduction number is less than 1, the disease free coexistence equilibrium is globally asymptotically stable. If the basic reproduction number is larger than 1, our system is uniformly persistent. The unique coexisting endemic disease equilibrium exists and is globally stable under certain conditions. However, infection of the inferior competitor does not change the outcome.