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| In this work we study some features of the behavior of Leslie type matrix models for age structured populations subjected to environmental stochasticity. In these models, the population lives in an environment that changes randomly with time in each projection interval. These changes can account for the variability in resources, climatic conditions, etc., that reflect in the fertility and survival coefficients os the population. In this way, there is a collection of Leslie matrices, each one of them corresponding to an environmental condition, and in each time step of the model the environment to which the population is subjected is defined by a certain random variable that is usually chosen to be a Markov chain.
The main parameter that controls the dynamics of these kind of models is the so called stochastic growth rate (s.g.r.). When the s.g.r is positive the population grows exponentially with probability one and when the s.g.r is negative the population goes extinct with probability one.
However, even in very simple situations, it is not possible to calculate the s.g.r. analytically. As a result, there are not simple expressions that biologists can use to check the extinction-explosion of populations.
In this work we build different upper and lower bounds for the s.g.r. that are tighter than the ones already existing in the literature. We analyze the conditions under which each bound works best. Finally, the different bounds are used to give necessary-sufficient conditions, based in simple expressions easy to check in practice, for the explosion and the extinction of the population. The general results are applied to the case of a population structured in juveniles and adults living in an ambient in which there two possible environments. |
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