Stochastic population models
In recent years, there has been an upsurge in interest for a certain class of
simple single-species population models, no doubt largely
because of their intriguing dynamical properties. They have provided the
archetypes of elementary chaos theory where they remain a source of
discovery and challenge.
These classical models meet the criteria set by Hassell who introduced
a systematic approach to the development of so-called density-dependent
population models in discrete time. These models take the
form
xt+1 = f(xt), t=0,1,2,...
Here xt
represents the population density at time t or in generation t.
The models should according to Hassell meet two fundamental criteria:
1. The population should have the potential to increase exponentially for small
populations, but
2. there should be a density-dependent feedback which reduces the actual rate
of increase as the population grows.
Such requirements lead us naturally to unimodular functions f increasing
from the origin up to some point c and decreasing to the right of c.
Examples of such models are the Ricker model
xt+1 = xtexp(r - g xt)
and the Hassell model
xt+1 = r xt / (1 + xt)b
where r (or exp(r)) is the intrinsic growth rate for small populations and
g and b, respectively, represent the inhibitive density-dependent feedback,
usually attributed to the environment.
The dynamical systems generated by unimodular functions f as above exhibit a
rich variety in their asymptotic behavior. In our project we want to
investigate some stochastic models, equally reasonable from the
``biologically logical'' point of view since they meet Hassell's two criteria.
Do they behave similarly or do new phenomena appear? The stochasticity
will be both demographic
(we model individuals and they do not necessarily multiply in exactly the same way) and environmental (the environmental
parameter gamma is allowed to vary, modeling, e. g., variable weather conditions). It turns out that the dynamics of the deterministic model is largely
preserved by demographic stochasticity whereas environmental stochasticity
may induce a growth-catastrophe behavior in the process .
For simplicity, we usually work with the Ricker model. Many of the results go through for other unimodular models as well.
At ÅAU, the research is carried out by Professor Högnäs and doctoral student Henrik Fagerholm.
Selected references:
Göran Högnäs: Quasi-stationary behaviour in a simple discrete-time model,
Åbo Akademi
Reports on Computer Science and Mathematics,
Ser.A, 2001. 10pp.
Henrik Fagerholm and Göran Högnäs:
Stability classification of a Ricker model with two random parameters,
Advances in Applied Probability 34 (2002), pp. 112 - 127.
Göran Högnäs: On the quasi-stationary distribution of a
stochastic Ricker model, Stoch. Proc. Appl. 70 (1997), pp.
243 - 263.
Mårten Ström: The influence of stochastic environmental
parameters on a stochastic population model [in Swedish], unpublished M.Sc. thesis, Åbo Akademi
University, Department of Mathematics 1998.
Michel Vellekoop and Göran Högnäs:
A unifying framework for chaos and stochastic stability in discrete
population models, J. Math. Biology
35 (1997), pp. 557 - 588.
Mats Gyllenberg, Göran Högnäs and Timo Koski:
Population models with environmental stochasticity, J.
Math. Biology 32 (1994), pp. 93 - 108.