Population Dynamics and Pattern Formation in an Info-chemical Mediated Tri-trophic Plankton Model

In this thesis, we study a spatio–temporal prey–predator model of plankton. This model has spatial interaction terms which represent a plankton dynamics that includes info–chemical mediated trophic interactions. We consider both a simplified two species model which has been studied in the literature (mostly in biological terms) and an extended, four-species model. In the latter, the grazing pressure of microzooplankton (M) on phytoplankton (P) is controlled through external infochemical (C) mediated predation by copepods (Z). We undertake a stability analysis of both the two species model and the four species model and compare the system dynamics. In relation to this, the critical conditions for Turing instability are derived; these are necessary and sufficient. Furthermore, we consider the degenerated situation wherein Turing bifurcation and Hopf bifurcation occur simultaneously. We also consider under what conditions Turing patterns are exhibited and under what conditions spatiotemporal patterns are observed generally. The Transient Turing instability of spatial interactions –exhibited by the two species model–is introduced and investigated in a number of ways. We also study the effects of the paradox of enrichment. This paradox led to a loss of stability in the four species model after this was derived from the two species model by expansion and by the addition of resources. Further, a numerical continuation technique was used to determine the existence of multiple stationary patterns

Stability of steady states of meta-food webs on discrete spatial networks

The concept of a food web is deceivingly simple. A simple map of interaction links between species. Nor is a spatially discrete network a particular daunting construct. Yet, even after almost a century of research there are still many unanswered questions about food webs and their spatial extensions, meta- food webs, and the perhaps most urgent one is, in the words of “father of modern ecology” (Slack, 2010), George Evelyn Hutchinson: “Why are there so many kinds of animals?” (Hutchinson, 1959) It has yet to be satisfyingly understood how complex food webs remain relatively stable and robust. The overwhelming complexity of real species relations and the difficulty for biologists and ecologists in gathering both precise and extensive field data makes it nearly impossible to faithfully recreate all nuances of actual food webs. This makes the topic particular appealing to the physicist who delights in abstracting problems to reveal underlying principles. The central focus of this thesis is thus to provide additional tools and insights to the topic of stability in meta-food webs. The generalized modelling method is particularly suited to this task as it is built around the idea of normalization to steady states which can be analysed concerning their stability. We offer an introduction to this method by examining the most simple food web possible consisting of a single predator and a single prey species. This provides a look at the fundamental terms and possibilities of the generalized modelling approach and gives some basic trends for the stability of food webs that are surprisingly sturdy in their applicability, e.g. the notion that large exponents for the primary production of biomass are destabilizing. We then add a spatial factor with a second patch so that we are dealing with a meta-food web. The food webs on each patch are homogeneous and we focus on the effect of migration between the two patches. Dispersal is overall destabilizing but can become less destabilizing for adaptive migration in certain parameter ranges. We also ask the question what dynamics occur during the transition from a stable to an unstable system which leads us to the phenomena that fall under the umbrella term of bifurcation. These simple systems show the full range of bifurcations including simple pattern building. From there we increase the complexity by incorporating heterogeneous food webs on each of the patches. This asymmetry allows for a wider range of behaviour at the point of bifurcation and now the additional element of synchrony between patches and species has to be taken into account. The ratio of oscillatory behaviour in case of perturbation increases and the oscillations becomes more anti- phasic compared to the homogeneous food webs; indicators of a higher robustness. The impact on linear stability cannot be easily predicted. We then extend the meta-food web from two patches and two species to many species on spatially distributed networks of patches though only with homogeneous local food webs. We show the analogy between reaction-diffusion systems on continuous space and on networks and how this can be applied to meta-food webs. Exploiting the inherent structure we can formulate a master stability function that allows for a separation of topological influences and those that stem from food web dynamics. Meta- communities become in general less stable for larger food webs and can be stable or unstable depending on the spatial configuration. They show primarily oscillatory and most likely rather localized responses to disturbances which are arguments for the robustness of the meta-communities. Finally, we summarize the results from the different sections. The steady states of food webs on spatial networks become less and less stable for increasing complexity but at the same time show signs of increasing robustness

Partial Differential Equations in Ecology

Partial differential equations (PDEs) have been used in theoretical ecology research for more than eighty years. Nowadays, along with a variety of different mathematical techniques, they remain as an efficient, widely used modelling framework; as a matter of fact, the range of PDE applications has even become broader. This volume presents a collection of case studies where applications range from bacterial systems to population dynamics of human riots

Nonlinear Dynamics of a Nutrient-Plankton Model

We investigated a nonlinear model of the interaction between nutrients and plankton, which was addressed using a pair of reaction-advection-diffusion equations. Based on numerical analysis, we studied a model without diffusion and sinking terms, and we found that the phytoplankton density (a stable state) increased with the increase of nutrient density. We analyzed the model using a linear analysis technique and found that the sinking of phytoplankton could affect the system. If the sinking velocity exceeded a certain critical value, the stable state became unstable and the wavelength of phytoplankton increased with the increase of sinking velocity. Furthermore, band patterns were also produced by our model, which was affected by the diffusion and sinking of phytoplankton. Thus, the change in the diffusion and sinking of phytoplankton led to different spatial distributions of phytoplankton. All of these results are expected to be useful in the study of plankton dynamics in aquatic ecosystems

Spatial modelling in plant ecology

In this thesis a range of lattice based spatially explicit models of ecosystems are presented and their applicability to various ecological situations is demonstrated with emphasis on plant communities These mechanistic and individual based models which include coupled map lattices and cellular automata aim to produce ecological insights and testable results Models of both short and long term systems are developed with the former being potentially testable in the eld and the latter promoting understanding where experimentation is not feasible A range of graphical and numerical techniques were developed to investigate both plant and animal model ecosystems The starting point is a short term single species coupled map lattice which investigates popula tion structure arising from local competitive interactions The model concludes that increase of size variation with increasing density indicates the presence of competitive intraspecic asymme try This idea is applied to crop data where considerable asymmetry is identied emphasising the need for balancing crop yield and size consistency Multispecies extensions to this model focus on spatial patterning arising from biotic interac tions and various numerical techniques underline the asymmetrical relationship between long and short lived species Environmental heterogeneity is imposed on the plant species in a third version of the model via the incorporation of an explicit resource base The complex inter dependence of community and environment is highlighted and illustrated by a model of the evolution of seed sizes Through the application of cellular automata to forest and epidemiological systems the concept of memory such as age or stage structuring is shown to be vital in the generation of spatial structure in long term ecological systems Analytical investigations generate further insights and again emphasise the crucial role played by spatial extensiveness in the wide range of ecological situations considered here In conclusion lattice models are ideally suited to the study of ecosystem