Intra-specific competition in population dynamics can be described by integro-differentialequations where the integral term corresponds to nonlocal consumption of resources by individualsof the same population. Already the single integro-differential equation can show theemergence of nonhomogeneous in space stationary structures and can be used to model the processof 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)]. Onthe other hand, competition of two different species represents a well known and well studiedmodel in population dynamics. In this work we study how the intra-specific competition can influencethe competition between species. We will prove the existence of travelling waves for the casewhere the support of the kernel of the integral is sufficiently narrow. Numerical simulations willbe carried out in the case of large supports.

A one-dimensional lattice of SIR (susceptible/infected/removed) epidemic centres isconsidered numerically and analytically. The limiting solutions describing the behaviour of thestandard SIR model with a small number of initially infected individuals are derived, and expressionsfound for the duration of an outbreak. We study a model for a weakly mixed populationdistributed between the interacting centres. The centres are modelled as SIR nodes with interactionbetween sites determined by a diffusion-type migration process. Under the assumption of fastmigration, a one-dimensional lattice of SIR nodes is studied numerically with deterministic andrandom 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. Explicitformulae are found for the speed of the travelling waves and compared with results of numericalsimulation. Approximate formulae for the epidemic propagation speed are also derived whencoupling coefficients are randomly distributed, they allow us to estimate how the average speed inrandom media is slowed down.

Basic models suitable to explain the epidemiology of dengue fever have previouslyshown the possibility of deterministically chaotic attractors, which might explain the observedfluctuations found in empiric outbreak data. However, the region of bifurcations and chaos requirestrong enhanced infectivity on secondary infection, motivated by experimental findings ofantibody-dependent-enhancement. Including temporary cross-immunity in such models, which iscommon knowledge among field researchers in dengue, we find bifurcations up to chaotic attractorsin much wider and also unexpected parameter regions of reduced infectivity on secondaryinfection, realistically describing more likely hospitalization on secondary infection when the viralload becomes high. The model shows Hopf bifurcations, symmetry breaking bifurcations of limitcycles, coexisting isolas, and two different possible routes to chaos, via the Feigenbaum perioddoubling and via torus bifurcations.

The challenges to live in the open water and the diversity of habitats in the marine environmentsprompts phytoplankton to devise strategies which often involve production of toxinsby 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 phenomenaand non-linear non-equilibrium pattern formation in a phytoplankton-zooplankton systemwith Holling type IV functional response. Type IV functional response yields to type II responsein 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 competitivesystems. Fish predation has a significant role in the temporal evolution of spatial patternsof phytoplankton-zooplankton system. From a field ecologist's perspective, it is not only importantto know the stable stationary patterns of an evolving phytoplankton-zooplankton system, butalso to know at what point of time these observations are carried out. This has implications forecological theory.

Nowadays there has been a growing consciousness among the tea industry to reducethe use of the chemical pesticides for pest control. Predators are beneficial insects that feed onharmful insects and mites, which incur considerable loss of production of tea. In this paper wehave considered a tritrophic model consisting of tea plant, pest and predator to analyze differentfield 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 biologicalcontrol.

Relaxation oscillations are limit cycles with two clearly differenttime scales. In this article the spatio-temporal dynamics of astandard prey-predator system in the parameter region of relaxationoscillation is investigated. Both prey and predator population aredistributed irregularly at a relatively high average level between amaximal and a minimal value. However, the slowly developing complexpattern exhibits a feature of “inverse excitability”: Bothpopulations show collapses which occur erratically both in space andin time. The nature of these collapses is analysed statistically andit is shown that the model behaviour can be interpreted as aresolution 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.