Abstracts of Posters 8-th European Conference on Mathematical ...

<strong>European</strong> <strong>Conference</strong> on Ma<strong>th</strong>ematical and Theoretical Biology 2011
Ma<strong>th</strong>ematical Models in Eco-epidemiology II; Wednesday, June 29, 11:00
Narcisa Apreutesei
Department <strong>of</strong> Ma<strong>th</strong>ematics, Technical University "Gh. Asachi" Iasi,
Romania
e-mail: napreut@gmail.com
Travelling wave solutions for integro-differential equations
from population dynamics
Our talk concerns some classes <strong>of</strong> integro-differential equations from population
dynamics, where <strong>th</strong>e integral term describes <strong>th</strong>e nonlocal consumption <strong>of</strong> resources.
Bo<strong>th</strong> monostable case and bistable case are investigated. Fredholm property <strong>of</strong> <strong>th</strong>e
corresponding linear operators can help to prove <strong>th</strong>e existence <strong>of</strong> travelling wave solutions.
For some models, we can prove <strong>th</strong>e existence <strong>of</strong> traveling waves only when
<strong>th</strong>e support <strong>of</strong> <strong>th</strong>e integral is sufficiently small. In <strong>th</strong>is case, <strong>th</strong>e integro-differential
operator is close to <strong>th</strong>e differential one. One uses a perturbation me<strong>th</strong>od which
combines <strong>th</strong>e Fredholm property <strong>of</strong> <strong>th</strong>e linearized operators and <strong>th</strong>e implicit function
<strong>th</strong>eorem. For large support, numerical simulations show <strong>th</strong>e propagation <strong>of</strong>
periodic travelling waves. For some o<strong>th</strong>er models, Leray-Schauder me<strong>th</strong>od can be
applied. This implies <strong>th</strong>e construction <strong>of</strong> a topological degree for <strong>th</strong>e corresponding
operators and <strong>th</strong>e establishment <strong>of</strong> a priori estimates for <strong>th</strong>e solution. Some
biological interpretations follow from <strong>th</strong>is study.
51