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Author(s): Lutz Becks [1, 5]; Frank M. Hilker [2]; Horst Malchow [2]; Klaus Jürgens [3, 4]; Hartmut Arndt (corresponding author) [1, 5] Discovering why natural population densities change over time [...]

The complexity and disorder in the spatiotemporal arrangement of aquatic environments, and the influence of biological systems on this arrangement are investigated. It is shown with the help of results that modeling by reaction-diffusion equations method is considered suitable to examine basic processes of intricate spatiotemporal fish and plankton dynamics.

Species competition in a fluctuating environment is often modelled with stochastic reaction-diffusion equations. In most cases the movement of individuals is described as Fickian diffusion. However, in heterogeneous environments this is not the first choice. Recently, it has been shown that Fokker-Planck diffusion describes the movement of species in a more realistic way. Fickian diffusion always leads to spatially uniform stationary distributions whereas the Fokker-Planck diffusion generates nonuniform solutions according to the heterogeneity of the environment and the corresponding spatial variation of diffusion. Species accumulate in regions of low diffusivity and tend to lower their densities in areas of high diffusivity. In the present paper, the impact of Fokker-Planck diffusion is studied with particular consideration of changing spatio-temporal population patterns during the competitive invasion of a spatially heterogeneous, populated habitat. The standard Lotka-Volterra competition model is applied to describe the resident-invader interaction. The resident is assumed to be adapted to the heterogeneous living conditions, i.e., its motion is modelled as space-dependent Fokker-Planck diffusion. The invader's diffusion is taken as neutral Fickian. Furthermore, it is shown that multiplicative environmental noise can either foster or hinder the invasion. [ABSTRACT FROM AUTHOR]

Semi-arid ecosystems made up of patterned vegetation, for instance, are thought to be highly sensitive. This highlights the importance of understanding the dynamics of the formation of vegetation patterns. The most renowned mathematical model describing such pattern formation consists of two partial differential equations and is often referred to as the Klausmeier model. This paper provides analytical and numerical investigations regarding the influence of different parameters, including the so-far not contemplated evaporation, on the long-term model results. Another focus is set on the influence of different initial conditions and on environmental noise, which has been added to the model. It is shown that patterning is beneficial for semi-arid ecosystems, that is, vegetation is present for a broader parameter range. Both parameter variability and environmental noise have only minor impacts on the model results. Increasing mortality has a high, nonlinear impact underlining the importance of further studies in order to gain a sufficient understanding allowing for suitable management strategies of this natural phenomenon. [ABSTRACT FROM AUTHOR]

Microscopic turbulent motions of water have been shown to influence the dynamics of microscopic species living in that habitat. The number, stability, and excitability of stationary states in a predator–prey model of plankton species can therefore change when the strength of turbulent motions varies. In a spatial system these microscopic turbulent motions are naturally of different strength and form a heterogeneous physical environment. Spatially neighboring plankton communities with different physical conditions can impact each other due to diffusive coupling. We show that local variations in the physical conditions can influence the global system in form of propagating pulses of high population densities. For this we consider three different local predator–prey models with different local responses to variations in the physical environment. The degree of spatial heterogeneity can, depending on the model, promote or reduce the number of propagating pulses, which can be interpreted as patchy plankton distributions and recurrent blooms. [ABSTRACT FROM AUTHOR]

Highlights: [•] We model modification of intraspecific prey competition by an infectious disease. [•] The effect on disease spread is investigated in an eco-epidemiological model. [•] Stable coexistence of pathogen and predator on shared prey/host becomes possible. [•] Bistability may lead to a prey population on the brink of disease-induced extinction. [•] We discuss possible implications for wildlife disease control programs. [ABSTRACT FROM AUTHOR]

In this paper, prompted by the practical issue of controlling the Aujeszky disease affecting hogs, we analyse what strategies can be implemented to try and contain an SI epidemic invasion in one and two neighbouring farms. We perform some simulations under different assumptions on the basic system, leading thus to different models. The results show that the disease cannot be arrested completely, since infected remain so for life and the system is of a diffusive type, but its spread can rather be slowed down. The influence of contamination islands, sudden external disease infiltration and containment barriers are simulated. The possible leaking of the infection into a neighbouring farm is also studied. The analysis is then extended to an SIS model allowing disease recovery. Under suitable conditions, harvesting is shown to be an effective containment strategy. [ABSTRACT FROM PUBLISHER]

A competition–diffusion model is considered for conditions of invasibility of a certain model area occupied by a native species. The instability of the environment leads to temporary extinction of both species at a randomly chosen time and spatial range. The spatiotemporal dimension of these extreme fragmentation events as well as a possible selected removal of the invader turn out to be the crucial driving force of the system dynamics. [ABSTRACT FROM AUTHOR]

We investigate the switching feeding behavior of predators in the context of one single prey population, which is disease affected. We consider the case of hunting indiscriminately both types of prey, when the infected prey causes no harm to their predators, but assume also in another model that feeding on the infected individuals has a negative return on the predators. Some counterintuitive results are obtained and discussed. [ABSTRACT FROM AUTHOR]

A model of a phytoplankton-zooplankton prey-predator system with viral infection of phytoplankton is investigated. Virus particles (V) are taken into account by an explicit equation. Phytoplankton is split into a susceptible (S) and an infected (I) class. A lytic infection is considered, thus, infected phytoplankton cells stop reproducing as soon as the infection starts and die at an increased mortality rate. Zooplankton (Z) is grazing on both susceptible and infected phytoplankton following a Holling-type II functional response. After the local dynamics of the V-S-I-Z system is analysed, numerical solutions of a stochastic reaction-diffusion model of the four species are presented. These show a spatial competition between zooplankton and viruses, although these two species are not explicitly coupled by the model equations. [ABSTRACT FROM AUTHOR]

Strange periodic attractors with complicated, long-lasting transient dynamics are found in a prey-predator model with disease transmission in the prey. The model describes viral infection of a phytoplankton population and grazing by zooplankton. The analysis of the three-dimensional system of ordinary differential equations yields several semi-trivial stationary states, among them two saddle-foci, and the sudden (dis-)appearance of a continuum of degenerated nontrivial equilibria. Along this continuum line, the equilibria undergo a fold-Hopf (zero-pair) bifurcation (also called zip bifurcation). The continuum only exists in the bifurcation point of the saddle-foci. Especially interesting is the emergence of strange periodic attractors, stabilizing themselves after a repeated torus-like oscillation. This form of coexistence is related to persistent and permanent ecological communities and to bursting phenomena. [ABSTRACT FROM AUTHOR]