Population dynamics with a refuge: fractal basins and the suppression of chaos

Study on Fractal Characteristics of Migration-Population Flow—Evidence from Egypt

Population migration is a major event of optimizing the allocation of production factors and a key way to construct regional relations and promote spatial reconstruction. However, there are few papers published on population migration that have a direct impact on the sustainable development of deserts owing to the more sensitive and complex man–earth relationship. Therefore, it is important to study the laws and characteristics of population migration in such regions. The study of Egypt by Zipf’s law shows that the spatial distribution of migration population size conforms to the law of power function with fractal characteristics of different types. The migration population is generally in a “Pareto” state in spatial distribution. Decentralization power is the leading driving force of spatial distribution, and scale-free regional distribution shows significant spatial agglomeration and gradient. Limited by research scope and “heavy tail” factors, Zipf’s law is not completely applicable. The spatial pattern and flow field characteristics of the migration population are analyzed in this paper by the conversion from “structural fractal” to “spatial fractal” based on scale-free geographic projection, providing the reference for the formulation of population governance policies and spatial planning strategies in Egypt and more desert countries.

Boundedness, persistence and stability for classes of forced difference equations arising in population ecology

Boundedness, persistence and stability properties are considered for a class of nonlinear, possibly infinite-dimensional, forced difference equations which arise in a number of ecological and biological contexts. The inclusion of forcing incorporates the effects of control actions (such as harvesting or breeding programmes), disturbances induced by seasonal or environmental variation, or migration. We provide sufficient conditions under which the states of these models are bounded and persistent uniformly with respect to the forcing terms. Under mild assumptions, the models under consideration naturally admit two equilibria when unforced: the origin and a unique non-zero equilibrium. We present sufficient conditions for the non-zero equilibrium to be stable in a sense which is strongly inspired by the input-to-state stability concept well-known in mathematical control theory. In particular, our stability concept incorporates the impact of potentially persistent forcing. Since the underlying state-space may be infinite dimensional, our framework enables treatment of so-called integral projection models (IPMs). The theory is applied to a number of examples from population dynamics.

ROLE OF PROTECTION IN A TRI-TROPHIC FOOD CHAIN DYNAMICS

In this paper, the role of protection in stabilizing the tri-trophic food chain dynamics has been explored. The density-dependent protection is provided to bottom prey or middle predator or both. It favors the oscillations damping and has the potential to control the chaotic fluctuations of population density. The bifurcation diagrams have been drawn with respect to protection parameter. They exhibit coexistence of all three species in the form of periodic solutions. The coexistence in the form of stable equilibrium is possible for higher values of protection parameters. Further increase in protection parameters may lead to extinction of one or two species. A two-parameter bifurcation diagram has also been drawn. The Poincaré Maps further confirm the role of protection in controlling the chaos.

Chaotic dynamics can select for long-term dormancy

Extended dormancy in a population is evolutionarily costly unless some variance in season-to-season fitness (usually driven by variance in environmental quality) makes bet hedging useful. Consequently, dormancy in a population is usually accepted as evidence of environmental variance. Using a Ricker-type model with heritable variation in dormancy, we show that this need not be so. Intrinsic population dynamics can generate chaotic fluctuations in the absence of environmental variance. Chaotic dynamics increase the frequency of a range of dormant strategists under natural selection, even when mortality during dormancy is relatively high. The buffering effect of dormant individuals then eliminates chaotic dynamics or generates periodic orbits of relatively low amplitude. These stabilized populations harbor a high frequency of dormant individuals that express a range of propensities to enter dormancy.