Print PageStage structured population models are the prevailing tool in conservation biology to analyze population viability and to predict the impact of different management actions. The population dynamics of many animal and plant species are affected by density dependent processes; knowing the local carrying capacity of animal and plant populations is particularly important in the face of increasingly fragmented landscapes. To estimate the asymptotic population size of particular habitats ecologists rely largely on simulations. We mathematically analyzed the asymptotic behavior of models with density dependent recruitment. We consider populations that are described by discrete-time stage-structured projection models. The population vector x(t) at time t is either finite dimensional if the stage structure is discrete (matrix model), or is in a function space if the stage structure is continuous (integral projection model). Furthermore we consider plant populations models with two types of seed bank: In the first case we assume that all seeds have the same germination and survival rates. In the second case we incorporate the dynamics of seeds within the soil and make germination and survival a function of seed depth.
Results/Conclusions
With many types of density dependent recruitment, we prove that there is an asymptotic population and stage structure that is independent of the nonzero initial population and stage structure. We give formulas for the asymptotic population density and quantity in terms of the life-history parameters, and provide the qualitative and ecological reasoning behind the proof. We identify those systems parameters for which the attractor is zero, parameters for which the attractor is positive, and those parameters for which there is no asymptotically stable equilibrium. We also consider a Ricker model, where the asymptotic behavior is more complicated. We give results for two models with a seed bank. We obtain asymptotic stability results for models with a scalar density-dependent seed bank, and with a structured density-independent seed bank. Furthermore, we analyze how sensitive the transient dynamics are to the choice of the functional form for the density dependence. We apply these results to a Platte Thistle (Cirsium canescens) model, a Blowout Penstemon (Penstemon haydenii) model, and a model of a hypothetical species with a seed bank.
