Palestrante
Descrição
The classical Lotka-Volterra equations are used to describe the dynamics of two species that interact in an ecosystem as predator and prey. Let the population of these species be denoted by y and x, respectively, so the model has the form: $\dot{x}= x(\alpha -\beta y)$, $\dot{y}=y(-\gamma +\delta x) $ with all coefficients being positive constants.Several generalizations of these equations are used to study the evolution of species population in ecosystems: more species can be considered, each one affecting the others; the functional response of predator to preys, defined as the predation rate divided by number of predators – in classical model it is $\beta x $ – can be a more complex function of the food density, and factors as changes in the environment or delays related to gestation time can be included.In nature, one can see that the overall interaction of species in the ecosystem often leads to stable dynamics, where populations oscillate or remain practically constant. Human beings may also induce systems to guarantee that one specie will not vanish from the environment, or become destructively numerous. Mathematical ecology hence requires the study of the conditions in which the ecological dynamical models are robust, i.e., small perturbations in the systems do not shift drastically its long-time behavior.In this work I show the results obtained by Tian, X et al and Forde, J.E. in the application of qualitative ordinary differential equations' theory to study the global dynamics of a Lotka-Volterra equation with Holling II functional response, presented in (1) and a predator-prey model with delay due to predator gestation (2), focusing in finding the conditions that guarantee limit cycles or stable equilibrium points, and the regions where stability holds.
Referências
1 TIAN, X et al. Global dynamics of a predator-prey system
with Holling type II functional response. Nonlinear Analysis: modeling and control, v.16, n.2, p.242-253, 2011.
2 FORDE, J.E. Delay differential equation models in
mathematical biology 2005.104p. Dissertation (Doctor of Philosophy) - The University of Michigan, Michigan, 2005.
| Apresentação do trabalho acadêmico para o público geral | Sim |
|---|---|
| Subárea | Física Aplicada |
