Kinesis In Animal Dispersal: Dependence Of Diffusion On Reproduction Coefficient

Migration and dispersal of animals has evolved under the control of natural selection. In a simple formalisation, the strategy of dispersal should increase Darwinian fitness. In this dissertation, we aim to introduce and analyse new models of purposeful kinesis with diffusion coefficient dependent on fitness. New models include one additional parameter, intensity of kinesis, and may be considered as the minimal models of purposeful kinesis. It is demonstrated how kinesis could be beneficial for assimilation of patches of food or of periodic fluctuations. Nevertheless, kinesis, based on local and instantaneous estimations of fitness, is not always beneficial in the long-term and spatially global perspective: for example, for species with the Allee effect it can delay invasion and spreading. It is proven that kinesis cannot modify stability of homogeneous positive steady states. In the proposed basic model, mobility decreases for high reproduction coefficient. Therefore, animals stay longer in good conditions and leave quicker bad conditions. Accounting for the cost of mobility in the reproduction ratio leads to an equation for mobility. It can be solved in a closed form using Lambert W-function. These models with the simple linear cost of mobility have an intrinsic phase transition: when conditions worsen then the mobility increases up to some critical value of the reproduction coefficient. For worse conditions, there is no solution for mobility. We interpret this bifurcation as the complete loss of mobility that is degeneration of diffusion. Mobility increases with worsening of conditions up to some limit, and after that, mobility nullifies. We analyse the impact of the purposeful kinesis model on running waves. Both monotonic and non-monotonic (the Allee effect) dependence of the reproduction coefficient on the population density are studied. The possible benefits of the purposeful kinesis are demonstrated: with the higher diffusion, while the population without kinesis ends up with extinction, the population with kinesis stays alive and has the running wave behaviour.