In this work, we examine the mathematical analysis and numerical simulation of pattern formation in a subdiffusive multicomponents fractional-reaction-diffusion system that models the spatial interrelationship between two preys and predator species. The major result is centered on the analysis of the system for linear stability. Analysis of the main model reflects that the dynamical system is locally and globally asymptotically stable. We propose some useful theorems based on the existence and permanence of the species to validate our theoretical findings. Reliable and efficient methods in space and time are formulated to handle any space fractional reaction-diffusion system. We numerically present the complexity of the dynamics that are theoretically discussed. The simulation results in one, two and three dimensions show some amazing scenarios.

A general framework for age-structured predator-prey systems is introduced. Individualsare distinguished into two classes, juveniles and adults, and several possibleinteractions are considered. The initial system of partial differential equations isreduced to a system of (neutral) delay differential equations with one or two delays.Thanks to this approach, physically correct models for predator-prey with delay areprovided. Previous models are considered and analysed in view of the above results. ARosenzweig-MacArthur model with delay is presented as an example.

Habitat fragmentation is an important area of concern in species conservation. Habitatfragmentation can affect population distributions through reductions in suitable habitat,and through organism responses to different habitat types and the transitions betweenthem. In earlier work, the effect of habitat fragmentation on cyclic populations wasinvestigated in the context of populations that show no behavioural response to theinterface between habitat types. In this paper, we extend the earlier work by addingedge-mediated behaviour to the models. That is, we investigate the dynamics that resultwhen oscillatory predator and prey species also exhibit behavioural responses to habitatinterfaces. Our results show generally that habitat loss decreases the amplitude and theaverage density of the prey and predator populations, but that most of the reponsesobserved in the two models exhibit marked differences. This work highlights the complexityof the interplay between population cycles, habitat fragmentation, and edge-mediatedbehaviour, and the need to study such systems in greater detail.

A minimal model for predator-prey interaction in a composite environment is presented and analysed. We first consider free migrations between two patches for both interacting populations, and then the particular cases where only one-directional migration is allowed and where only one of the two populations can migrate. Our findings indicate that in all cases the ecosystem can never disappear entirely, under the model assumptions. The predator-free equilibrium and the coexistence of all populations are found to be the only feasible stable equilibria. When there are only one-directional migrations, the abandoned patch cannot be repopulated. Other equilibria then arise, with only prey in the second patch, coexistence in the second patch, or prey in both patches but predators only in the second one. For the case of sedentary prey, with predator migration, the prey cannot thrive alone in either of the two environments. However, predators can survive in a prey-free patch due to their ability to migrate into the other patch, provided prey is present there. If only the prey can migrate, the predators may be eliminated from one patch or from both. In the first case, the patch where there are no predators acts as a refuge for the survival of the prey.

Although herbivores may account for a significant source of seed and seedling mortality in many tropical tree species, plant species differ in their response to seed damage. Here we investigate the relative effects of seed predation on the regeneration of five tree species in a mid-elevation Papua New Guinean rain forest. Exclosure treatments and shade-house experiments were monitored from November 2004 to March 2006 to assess the differential effects of seed predation on seed viability and seedling growth. Results indicate that although seed predators attack all five focal species, they influence the seedling populations in two, Cerbera floribunda and Microcos grandiflora, and minimally affect the seedling populations of Terminalia impediens, Pandanus penicillus and Endiandra latifolia in the years measured. Predation and germination frequencies were compared to the abundance of focal species at several life stage classes to explore potential correlations between species-specific seed mortality patterns and life stage distributions. We found that the species-specific influence of mammalian seed predators correlated with abundance distributions in three life stages. Species with high survivorship after seed predator attacks displayed a significant decrease in abundance from the seedling-to-sapling transition, while those species with high seed mortality demonstrated relatively even distributions of seedlings, saplings and adults (> 10 cm dbh). These contrasting patterns suggest that differential seed predation effects on regeneration may play a key role in the recruitment of individuals to the sapling stage.

The effects of predators on prey populations can be modified by a number of abiotic factors. Here, we investigated the combined and separate effects of rain and ground-dwelling predators on aphid populations in a microcosm experiment lasting for 21 days, using PCR to analyse the gut content of the predators. Rain significantly dislodged aphids from shoots and ears by 57% and 25%, respectively. The gut content analysis showed that more predators consumed aphids in the rain treatment than without rain, indicating higher availability of aphids to ground-dwelling predators after rain. However, no synergistic effects of rain and ground-dwelling predators on aphid population development could be demonstrated. Rain alone significantly decreased aphid populations by 27%, suggesting that this is a significant mortality factor. Predators alone had no significant effect on aphid numbers, but the gut content analyses showed aphid consumption also in the no-rain treatments, indicating that aphids were available to the predators on the soil surface even without rain. Our results suggest that weather conditions such as rain can modify predator-prey interactions in the field. Employing PCR-based predator gut content analyses proved to be useful as trophic links could be directly verified.

A survey is presented of stochastic and deterministic developments in the study of the effects of nearest-neighbour ‘migration’ between spatially separated ‘colonies’. Such processes are of general applicability, and are relevant to any vector process X(t) = (X1(t), · ··, XN(t)) in which the arrival, departure and transfer rates for the states {X(t) = n} may be written in the form α i(ni), βi(ni) and γ ij(ni, nj), respectively, where n = (n1, · ··, nN). Whilst the main body of results are described in terms of birth-death, genetic and epidemic situations, the final section examines within colony interaction in the context of spatial predator-prey processes.