Miriam Sosa-Díaz and Faustino Sánchez-Garduño
A Qualitative Description of the Dynamics in the Interactions between Three Populations Pollinators, Plants, and Herbivores
307 - 324
2016
10
6
International Journal of Mathematical and Computational Sciences
https://publications.waset.org/pdf/10004763
https://publications.waset.org/vol/114
World Academy of Science, Engineering and Technology
In population dynamics the study of both, the
abundance and the spatial distribution of the populations in a
given habitat, is a fundamental issue a From ecological point of
view, the determination of the factors influencing such changes
involves important problems. In this paper a mathematical model to
describe the temporal dynamic and the spatiotemporal dynamic of the
interaction of three populations (pollinators, plants and herbivores) is
presented. The study we present is carried out by stages 1. The
temporal dynamics and 2. The spatiotemporal dynamics. In turn,
each of these stages is developed by considering three cases which
correspond to the dynamics of each type of interaction. For instance,
for stage 1, we consider three ODE nonlinear systems describing
the pollinatorplant, plantherbivore and plantpollinatorherbivore,
interactions, respectively. In each of these systems different types of
dynamical behaviors are reported. Namely, transcritical and pitchfork
bifurcations, existence of a limit cycle, existence of a heteroclinic
orbit, etc. For the spatiotemporal dynamics of the two mathematical
models a novel factor are introduced. This consists in considering
that both, the pollinators and the herbivores, move towards those
places of the habitat where the plant population density is high.
In mathematical terms, this means that the diffusive part of the
pollinators and herbivores equations depend on the plant population
density. The analysis of this part is presented by considering pairs of
populations, i. e., the pollinatorplant and plantherbivore interactions
and at the end the two mathematical model is presented, these models
consist of two coupled nonlinear partial differential equations of
reactiondiffusion type. These are defined on a rectangular domain
with the homogeneous Neumann boundary conditions. We focused
in the role played by the density dependent diffusion term into
the coexistence of the populations. For both, the temporal and
spatiotemporal dynamics, a several of numerical simulations are
included.
Open Science Index 114, 2016