Intra-specific competition in population dynamics can be described by integro-differential equations where the integral term corresponds to nonlocal consumption of resources by individuals of the same population. Already the single integro-differential equation can show the emergence of nonhomogeneous in space stationary structures and can be used to model the process of speciation, in particular, the emergence of biological species during evolution [S. Genieys et al., Math. Model. Nat. Phenom. 1 (2006), no. 1, 65-82], [S. Genieys et al., Comptes Rendus Biologies, 329 (11), 876-879 (2006)]. On the other hand, competition of two different species represents a well known and well studied model in population dynamics. In this work we study how the intra-specific competition can influence the competition between species. We will prove the existence of travelling waves for the case where the support of the kernel of the integral is sufficiently narrow. Numerical simulations will be carried out in the case of large supports.

A one-dimensional lattice of SIR (susceptible/infected/removed) epidemic centres is considered numerically and analytically. The limiting solutions describing the behaviour of the standard SIR model with a small number of initially infected individuals are derived, and expressions found for the duration of an outbreak. We study a model for a weakly mixed population distributed between the interacting centres. The centres are modelled as SIR nodes with interaction between sites determined by a diffusion-type migration process. Under the assumption of fast migration, a one-dimensional lattice of SIR nodes is studied numerically with deterministic and random coupling, and travelling wave-like solutions are found in both cases. For weak coupling, the main part of the travelling wave is well approximated by the limiting SIR solution. Explicit formulae are found for the speed of the travelling waves and compared with results of numerical simulation. Approximate formulae for the epidemic propagation speed are also derived when coupling coefficients are randomly distributed, they allow us to estimate how the average speed in random media is slowed down.

Basic models suitable to explain the epidemiology of dengue fever have previously shown the possibility of deterministically chaotic attractors, which might explain the observed fluctuations found in empiric outbreak data. However, the region of bifurcations and chaos require strong enhanced infectivity on secondary infection, motivated by experimental findings of antibody-dependent-enhancement. Including temporary cross-immunity in such models, which is common knowledge among field researchers in dengue, we find bifurcations up to chaotic attractors in much wider and also unexpected parameter regions of reduced infectivity on secondary infection, realistically describing more likely hospitalization on secondary infection when the viral load becomes high. The model shows Hopf bifurcations, symmetry breaking bifurcations of limit cycles, coexisting isolas, and two different possible routes to chaos, via the Feigenbaum period doubling and via torus bifurcations.

The challenges to live in the open water and the diversity of habitats in the marine environments prompts phytoplankton to devise strategies which often involve production of toxins by Harmful Algal Bloom (HAB) and rapid production of metabolites from non-toxic precursor. The functional response of the predator is described by Holling type IV. We investigate wave phenomena and non-linear non-equilibrium pattern formation in a phytoplankton-zooplankton system with Holling type IV functional response. Type IV functional response yields to type II response in the event of large immunity from or tolerance of prey. It has been found that the Wave of Chaos (WoC) is still effective mechanism for the propagation of chaotic dynamics in predation and competitive systems. Fish predation has a significant role in the temporal evolution of spatial patterns of phytoplankton-zooplankton system. From a field ecologist's perspective, it is not only important to know the stable stationary patterns of an evolving phytoplankton-zooplankton system, but also to know at what point of time these observations are carried out. This has implications for ecological theory.

Nowadays there has been a growing consciousness among the tea industry to reduce the use of the chemical pesticides for pest control. Predators are beneficial insects that feed on harmful insects and mites, which incur considerable loss of production of tea. In this paper we have considered a tritrophic model consisting of tea plant, pest and predator to analyze different field observations. The effect of discrete time-delay on the tritrophic model is studied critically. The dynamical behaviours are studied both analytically and numerically by computer simulation. Our results suggest some theoretical measures which may be implemented for successful biological control.

Relaxation oscillations are limit cycles with two clearly different time scales. In this article the spatio-temporal dynamics of a standard prey-predator system in the parameter region of relaxation oscillation is investigated. Both prey and predator population are distributed irregularly at a relatively high average level between a maximal and a minimal value. However, the slowly developing complex pattern exhibits a feature of “inverse excitability”: Both populations show collapses which occur erratically both in space and in time. The nature of these collapses is analysed statistically and it is shown that the model behaviour can be interpreted as a resolution of the paradox of enrichment.

We consider the number of trophic levels in a food chain given by the equilibrium state for a simple mathematical model with ordinary differential equations which govern the temporal variation of the energy reserve in each trophic level. When a new trophic level invades over the top of the chain, the chain could lengthen by one trophic level. We can derive the condition that such lengthening could occur, and prove that the possibly longest chain is globally stable. In some specific cases, we find that the possibly longest chain is such that the lower trophic level has a greater energy reserve than the higher has, so that the distribution of energy reserves can be regarded to have a pyramid shape, whereas, if any of its trophic levels is removed, the pyramid shape cannot be maintained. Further, we find the condition that arbitrary long chain can be established. In such unbounded case, we prove that any chain could not have the pyramid shape of energy reserve distribution.