### A new class of non-linear stochastic population models with mass conservation

S.A.L.M. Kooijman, J. Grasman and B.W. Kooi 2007.
A new class of non-linear stochastic population models with mass conservation.
*Math Biosci* **210**: 378 - 394
#### Abstract

We study the effects of random feeding, growing and dying
in a closed nutrient-limited producer/consumer system, in which
nutrient is fully conserved, not only in the mean, but also across
random events. More specifically, we relate these random effects to
the closest deterministic models, and evaluate the importance of the
various times scales that are involved. These stochastic models
differ from deterministic ones not only in stochasticity, but they
also have more detail because they don't separate time scales, while
the deterministic equivalents do. We tried to separate the effects
of more detail from that of stochasticity. The producers have
(nutrient) reserve and structure, and so a variable chemical
composition. The consumers have only structure, so a constant
chemical composition. The conversion efficiency from producer to
consumer, therefore, varies. This class of models is new, as far as
we know. The producers follow Droop's kinetics with fast nutrient
uptake and the consumers Monod's kinetics. The consumers use reserve
and structure of the producers as complementary compounds, following
the rules of Dynamic Energy Budget theory. Consumers die at constant
specific rate and decompose instantaneously. Stochasticity is
incorporated in the behaviour of the consumers, where the switches
to handling and searching, as well as dying are Poissonian point
events. We show that the stochastic model has one parameter more
than the deterministic formulation without time scale separation for
conversions between searching and handling consumers, which itself has
one parameter more than the deterministic formulation with time
scale separation for these conversions. These extra parameters are
the contributions of a single individual producer and consumer to
their densities, and the ratio of the two, respectively. The
tendency to oscillate increases with the number of parameters. The
focus bifurcation point has more relevance for the asymptotic
behaviour of the stochastic model than the Hopf bifurcation point,
since a randomly perturbed damped oscillation exhibits a behavior
similar to that of the stochastic limit cycle particularly near this
bifurcation point. For total nutrient values below the focus
bifurcation point, the system gradually becomes more confined to the
direct neighbourhood of the isocline for which the producers do not
change.