Effect of population size in a Prey-Predator model
Event details
| Date | 01.03.2013 |
| Hour | 10:15 › 11:00 |
| Speaker |
Claude Lobry EPI Modemic, Inra-Inria, 2 place Viala, Montpellier. |
| Location |
ME C2 405
|
| Category | Conferences - Seminars |
[Please note that this seminar will be in French]
In population dynamics state variables represent the size of some population and, by the way, are ``integers".
Birth and death processes and similar stochastic models are suitable models to represent discrete populations.
Nevertheless it is widely recognized that, when populations are large, thanks to the central limit theorem, ordinary
differential equations with continuous state variables are suitable approximations. But what means ``large " ?
We consider a stochastic version of the basic predator-prey differential equation model of Rosenzweig-MacArthur.
Our model contains a parameter ω which can be interpreted as the number of individuals for one unit of prey.
This means that if x denotes the quantity of prey in the differential equation model, x = 1 means that there are ω individuals in the discontinuous model.
It is shown by the mean of simulations and explained by a mathematical analysis based on results in singular perturbation theory (the so called theory of Canards) that qualitative properties of the model like persistence or extinction are dramatically sensitive to ω. For instance, in our example, if ω = 10^7 the model predicts extinction and if ω = 10^8 it predicts persistence. This means that we must be very cautious when we use continuous variables in place of jump processes in dynamic population modeling even when we use stochastic differential equations in place of a deterministic ones.
In population dynamics state variables represent the size of some population and, by the way, are ``integers".
Birth and death processes and similar stochastic models are suitable models to represent discrete populations.
Nevertheless it is widely recognized that, when populations are large, thanks to the central limit theorem, ordinary
differential equations with continuous state variables are suitable approximations. But what means ``large " ?
We consider a stochastic version of the basic predator-prey differential equation model of Rosenzweig-MacArthur.
Our model contains a parameter ω which can be interpreted as the number of individuals for one unit of prey.
This means that if x denotes the quantity of prey in the differential equation model, x = 1 means that there are ω individuals in the discontinuous model.
It is shown by the mean of simulations and explained by a mathematical analysis based on results in singular perturbation theory (the so called theory of Canards) that qualitative properties of the model like persistence or extinction are dramatically sensitive to ω. For instance, in our example, if ω = 10^7 the model predicts extinction and if ω = 10^8 it predicts persistence. This means that we must be very cautious when we use continuous variables in place of jump processes in dynamic population modeling even when we use stochastic differential equations in place of a deterministic ones.
Practical information
- General public
- Free
Organizer
- Colin Jones