Analysis of a time-dependent problem of mixed migration and population dynamics

In this work, we consider a system of differential equations modeling the dynamics of some populations of preys and predators, moving in space according to rapidly oscillating time-dependent transport terms, and interacting with each other through a Lotka-Volterra term. These two contributions naturally induce two separated time-scales in the problem. A generalized center manifold theorem is derived to handle the situation where the linear terms are depending on the fast time in a periodic way. The resulting equations are then amenable to averaging methods. As a product of these combined techniques, one obtains an autonomous differential system in reduced dimension whose dynamics can be analyzed in a much simpler way as compared to original equations. Strikingly enough, this system is of Lotka-Volterra form with modified coefficients. Besides, a higher order perturbation analysis allows to show that the oscillations on the original model destabilize the cycles of the averaged Volterra system in a way that can be explicitely computed

Process-dependence of biogenic feedback effects in models of plankton dynamics.

The prospect of human-induced climate change has stimulated research into several biological processes that might affect climate. One such process that has attracted a substantial research effort is the so-called CLAW hypothesis (Charlson et al. 1987). This hypothesis suggests that marine plankton ecosystems may effectively regulate climate by a feedback associated with the production of dimethylsulphide (DMS). Charlson et al. (1987) observed that some of the DMS produced by marine ecosystems is transferred from the ocean to the atmosphere where it is the major source of cloud condensing nuclei (CCN) over the remote oceans. The aerosols resulting from biogenic DMS emissions can have a direct effect on the solar radiative forcing experienced by the Earth through scattering, absorption and reflection and can also lead to increased cloud formation; the CLAW hypothesis proposes that these mechanisms could regulate climate. Charlson et al (1987) argued that an increase in global temperature would lead to increased biogenic DMS emissions from the ocean and result in an increase in scattering, cloud cover and cloud albedo that would increase the proportion of the incoming solar radiation reflected back into space (thus changing the global albedo), and thereby cooling the planet. The objective of this paper is to examine the implications of the climate regulation process proposed by Charlson et al. (1987) for the dynamics of the ecosystems that produce it. Cropp et al. (2007) developed a simple plankton model that incorporated the DMS feedback mechanism and compared its dynamics to the same ecosystem model without the feedback. These simulations revealed that the presence of the feedback generally enhanced the stability of the ecosystem by making it more resilient to perturbation. In this research, we compare the effect of the feedbacks on a similar NPZ ecosystem model that has a greater range of dynamical behaviour than the model used by Cropp et al. (2007). The results of simulations with the new feedback model are compared to the results of Cropp et al. (2007) to elucidate the influence of the model formulation on the effects of the feedback

A fractional B-spline collocation method for the numerical solution of fractional predator-prey models

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost

Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation

In this paper, we consider a non-autonomous stochastic Lotka-Volterra competitive system dxi(t) = xi(t)[(bi(t)¡ nPj=1aij (t)xj (t))dt+¾i(t)dBi(t)], where Bi(t) (i = 1; 2; ¢ ¢ ¢ ; n) are independent standard Brownian motions. Some dynamical properties are discussed and the su±cient conditions for the existence of global positive solutions, stochastic permanence, extinction as well as global attractivity are obtained. In addition, the limit of the average in time of the sample paths of solutions is estimated