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Spatial domain; introduced species enter the semicircular domains located on the left side (in smaller density for the predator population).
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The aim of this work is to develop and analyse a mathematical model for a predator-2 preys system arising in insular environments. We are interested in the evolution of a native prey population without behavioural traits to cope with predation or competition, after the introduction of alien species. Here, we consider a long living bird population w...
Contexts in source publication
Context 1
... Figure 3, W, the spatial domain is the unit square in R 2 , split into three subdomains: an in- ner zone, an outer zone and an intermediate one. ...
Context 2
... our numerical experiments we use the set of parameters depicted in Tables 1 and 2 corresponding to the duration of the adult stage. Spatial heterogeneity is charac- terised by different values for some parameters in the different zones in Figure 3. For the native prey population, favourable zones are described by low logistic coefficients k a and k j , unfavour- able ones by higher logistic coefficients. ...
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Citations
... In predator-prey interactions, many predators can distinguish the characteristics of different prey to hunt the most beneficial prey and avoid ingesting detrimental prey [1][2][3]. This phenomenon is widespread in terrestrial and aquatic ecosystems (especially in plankton ecosystems), which is termed as selective grazing behavior, and has attracted extensive attention in recent years [4,5]. ...
In order to explore the pattern formation of non-toxic phytoplankton, toxin-producing phytoplankton, and zooplankton, a spatiotemporal plankton model is formulated by incorporating the component Allee effect, additional food, and selective grazing. For the non-spatial scenario, the qualitative analyses and numerical simulations show that the system admits rich dynamics such as Hopf bifurcation and alternative stable states. For the spatial model, the stability analyses demonstrate that the moderate component Allee effect and adequate additional food promote stable coexistence. Furthermore, our findings illustrate that selective grazing facilitates toxin-producing phytoplankton biomass. Low grazing selectivity decreases zooplankton biomass. Moderate grazing selectivity contributes to coexistence. Excessive grazing selectivity promotes the rapid reproduction of toxin-producing phytoplankton and leads to spatiotemporal chaos. In addition, the variations in the component Allee effect, additional food, and grazing selectivity give rise to the transitions of spatial patterns, which shed valuable insights into controlling plankton distribution.
... For example, oscillations can be introduced into the ecosystem or they can be controlled and eliminated by an appropriate choice of additional food [60]. The so-called paradox of enrichment, permanence and persistence of systems can always be achieved via a suitable choice (depending on numerous parameters of the ecosystem) of additional food [14,12]. The type of additional food provided to predators can help eliminate the prey (right food provided to predators), or eliminate the predators by distracting result of additional food to predators is crucial in this study. ...
We formulate a robust mathematical model for two predators having an overlapping dietary niche breadth. We use the Beddington–DeAngelis functional and numerical responses which are relevant in addressing the principle of competitive exclusion as species interact. The stabilizing effect of additional food in relation to the relative diffusivity D, and wave number k, has been investigated. Stability, dissipativity, permanance, persistence and periodicity of the model have been studied using the routine and limit cycle perturbation methods. Besides revealing irregular periodic travelling wave behaviour due to predator interference, numerical results also show oscillatory temporal dynamics resulting from additional food supplements combined with high predation rates.
... Without considering the mediation by the predator, Seabirds the two prey species may appear to be competing, hence the term apparent competition . Th e dynamics of apparent competition on islands have been theoretically studied for the specifi c case where the primary prey is a native species, and the predator and secondary prey are introduced mammals (Courchamp et al. 1999, Gaucel and Pontier 2005, Zhang et al. 2006 ). An introduced prey species evolutionarily adapted to mammalian predation via a high growth rate can facilitate an increase in abundance of the introduced predator. ...
This chapter focuses on the indirect effects of introduced predators on other animal species in seabird islands. It studies how introduced predators alter the outcomes of trophic and competitive interactions, and their effects within island food webs. It highlights the risks of eradicating seabird predators without identifying or managing the consequences of releasing other potentially powerful predators from control, and describes their role that varies greatly on the composition of vulnerable species and the island biogeography driving population dynamics. It also enumerates five possible predator-related indirect effects that depend on the trophic position of each species and the interaction between themselves: interspecific competition, apparent competition, intraguild predation, mutualisms, and multiple species.
... Bonnaud et al., 2007; Peck et al., 2008). The dynamics of such single predator models have been explored elsewhere, and also lead to prey extinction attributable to the superpredator (Gaucel and Pontier, 2005). We used a simple predator functional response for our model, assuming no prey preference beyond numerical availability. ...
a b s t r a c t Complete extirpation of a species can generate cascading effects throughout an ecosystem, yet are pre-cisely the goal of island eradications of pest species. "Mesopredator release effect", an asymmetrical special case of intraguild predation, has been hypothesised as a possible indirect effect from eradications, where superpredator removal can generate a mesopredator increase which may increase the impact on their shared prey. Theoretically this suggests that for intraguild predators, the superpredator may pro-tect the shared prey from mesopredation, and removal of superpredators alone is not recommended. We create a model of long-lived age-structured shared prey and explore the non-equilibrium dynamics of this system. The superpredator can impact all prey life-stages (adult survival and reproductive suc-cess) whereas the smaller mesopredator can only impact early life-stages (reproductive success). This model is independently tested with data from a closed oceanic island system where eradication of intro-duced intraguild predators is possible for conservation of threatened birds. Mesopredator release only occurs in strongly top-down moderated (resource-abundant) systems. Even when mesopredator release can occur, the negative impact of more mesopredators is outweighed by the benefit of superpredator removal, allowing recovery of the prey population. Results are robust to 10% variation in model parame-ters. The consideration of age-structured prey contradicts previous theoretical results for mesopredator release effect and intraguild predation. Superpredator eradication is vital for population recovery of long-lived insular species. Nonetheless island conservation must retain a whole-ecosystem perspective given the complex trophic relationships among multiple species on islands.
... • More complex systems. Three and four component systems based on [6] are found in [7], [9], [10] and [11]. Partial analytical results are supplied for the ODE systems but little is known for the PDE systems. ...
... • Spatial heterogeneities. The case of heterogeneous spatial environments is of interest: see [1], [3] and [15] for standard predator-prey systems and numerical simulations of the model under consideration in [9], [10] and [11]. ...
This note is dedicated to the question of global existence for solutions to a two component singular system of reaction-diffusion equations modeling predator-prey interactions in insular environments. Depending on a 2D parameter space, positive orbits of the underlying ODE system undergo interesting dynamics, e.g., finite time existence and global existence may coexist. These results are partially extended to the reaction-diffusion system in the case of identical diffusivities. Our analysis relies on an auxiliary non singular reaction-diffusion system whose solutions may or may not blow up in finite time. Numerical simulations illustrate our analysis, including a numerical evidence of spatio-temporal oscillations.
... To conclude, the present study concurs with previous empirical and theoretical evidence (see, e.g. Courchamp et al. 1999;Zavaleta et al. 2001;Fan et al. 2005;Frenot et al. 2005;Gaucel and Pontier 2005) suggesting that trophic relationships between cats and their native and introduced prey can be complex. Although the impacts ship rats, mice and rabbits have on the New Island ecosystem have not been fully quantiWed, it seems possible that cats, by limiting the numbers of their mammal prey, could play a positive role in the maintenance of local biodiversity. ...
We studied the diet of feral cats (Felis catus) on New Island, Falkland Islands, through the analysis of 373 scats collected during the austral summers of 2004/2005 and
2005/2006. The most frequent prey were three introduced mammals (house mice Mus musculus, ship rats Rattus rattus and rabbits Sylvilagus sp.) and the thin-billed prion Pachyptila belcheri (each season present on ca. 21% of the analysed scats). These represent the first systematic data on feral cat diet for the
Falklands. A simple bioenergetics model suggests that cats could be eating in the region of 1,500–11,000prions per season,
representing <1% of the local adult and subadult population. Predation on other seabirds nesting on New Island (several penguin
species, albatrosses and cormorants) was unimportant, with the possible exception of white-chinned petrels Procellaria aequinoctialis, which nest locally in very small numbers. For each prion eaten, cats were estimated to have killed 1.1–1.9 ship rats during
the summer season, and probably more in autumn and winter. Knowing that ship rats are prion predators, it is conceivable that,
on the whole, cats are having a positive impact on the prion population, a scenario predicted by general theoretical models.
Thus, considering the available information, we would not recommend the implementation of any eradication programme on New
Island that would target cats in isolation. Nevertheless, it would be prudent to consider some local action targeting cats
and rats around the small New Island white-chinned petrel colony.
... Cats have been introduced worldwide into a great number of continental and insular systems (Courchamp et al. 2003;Nogales et al. 2004). Feral cat predation has had widespread and detrimental effects on native fauna, especially on islands (Courchamp et al. 2003;Gaucel and Pontier 2005;Whittaker 1998). Feral cats consume a wide variety of insular fauna ranging from native arthropods, reptiles, and birds, to alien lagomorphs (Biro et al. 2005;Fitzgerald et al. 1991;Nogales and Medina 1996;Paltridge et al. 1997;Tidemann et al. 1994;Van Aarde 1980). ...
Predation by feral cats (Felis catus) is recognized as a major threat to native fauna worldwide, but the competitive effects of cats on native species have not been extensively studied. Cats occur on San Clemente Island, California, in sympatry with endemic island foxes (Urocyon littoralis clementae). We examined diets of cats and island foxes between years, seasons, and habitats to assess the potential for resource competition between the 2 species. Analysis of 602 cat and 958 fox feces revealed a high level of dietary overlap (O = 0.93) and relatively narrow niche breadths for both species (Bstandard Fox = 0.37; Bstandard Cat = 0.49). Despite the overlap in diet, cats and foxes appear to partition prey resources. Cats consume approximately equal proportions of arthropod (47.9%) and vertebrate (44.2%) prey, the latter primarily rodents (29.2%) and lizards (12.9%). In contrast, foxes appear to rely more on arthropods (57.7%), with plants (20.5%) and vertebrates (21.6%) occurring in lower, but roughly equal frequencies. Season appeared to have little effect on diet; however, diet did vary between habitats and years for both species. Diets of cats on San Clemente Island are consistent with those from other studies. We found no evidence of a dietary shift by foxes that were in sympatry with cats.
... Soient α et γ les taux de préférence respectifs des prédateurs pour les proies introduites et pour les proies natives, les termes de prédations deviennent αA α(γJ+A)+R pour les proies natives adultes, γαJ α(γJ+A)+R pour les proies natives juveniles et R α(A+J)+R pour les proies introduites. Le système s'écrit donc, cf [37]. ...
... Soit Ω le domaine spatial d'étude, on considère que tous les paramètres démographiques définis dans la section précédente sont maintenant des fonctions de la variable spatiale x. Le modèle (6) se dérive donc en un modèle de réaction diffusion (13), cf [35], [36] et [37]. ...
... Dans le même esprit, le modèle spatial pour le système JAC s'écrit sous la forme (16), cf [37]. ...
The aim of this thesis is to develop and analyse deterministic predator-prey models for species living in heterogeneous insular environments. We are interested in the evolution of a native prey population, after the introduction of alien species, predators and competitors. In a first part we look at the spatially unstructured models; this yields singular systems of ODEs, some denominator of the RHS can be zero. The mathematical analysis gives some conditions for persistence or finite time extinction of populations. The asymptotical behaviour depends on additional hypotheses. In a similar way, we study the case of a native prey species split into 2 age stages: juveniles and adults. The second part deals with spatial models. We derive models taking into account the spatial heterogeneities of the environment and their effects on demographic parameters. We use reaction-diffusion systems with a singular logistic right hand side. Detailed analysis of these models gives criteria for global existence versus finite time existence of the solutions. At the same time, we develop a well adapted numerical method, using splitting methods, to validate the spatial models and allow the study of invasion processes. Numerical results point out the essential role played by the introduced prey population in successful invasion of isolated native preys colonies by the predator species. Finally, the discrete age structure for the native species allows us to exhibit oscillatory behaviours.
Cette thèse a pour objet la modélisation de systèmes multi-proies–prédateurs. Elle s’intéresse particulièrement à l’influence du comportement d’un prédateur sur les interactions indirectes entre ses proies, i.e. l’effet de l’ajout d’une proie sur la densité des autres. La théorie classique prédit l’occurrence d’effets indirects négatifs entre les proies, ou compétition apparente, résultant de l’interaction avec un prédateur commun ; des résultats plus récents identifient certains mécanismes à même d’atténuer ces effets négatifs. Nos travaux revisitent les hypothèses autour de ces mécanismes dans des systèmes composés de deux proies et de leur prédateur commun. Après avoir fixé le cadre écologique en rappelant les principaux types d’interactions directes et indirectes, nous introduisons les modèles proies-prédateurs classiques. Les travaux se concentrent ensuite sur une famille de modèles présentant de la densité-dépendance négative chez les prédateurs couplée à différentes modélisations du comportement des prédateurs lorsqu’ils sont confrontés à plusieurs types de proies. Nous montrons notamment que les interactions entre ces mécanismes peuvent inverser la compétition apparente et, contre-intuitivement, accroître la densité des proies par l’intermédiaire d’un prédateur commun. Nos résultats trouvent pour partie application en lutte biologique, où il est courant de chercher à favoriser les auxiliaires en aménageant leur environnement (apport de nourriture alternative, refuge, ...). Ils suggèrent que de telles pratiques peuvent se révéler contre-productives, le contrôle des ravageurs pouvant être affaibli du fait d’une distraction de leurs prédateurs.
Invasive mammals have devastated endemic island communities throughout the world, and seabirds have proven particularly vulnerable,with many species extinctions. Introduced predators have had the greatest effect through direct predation, but this effect can now be mitigated by modern eradication techniques. However, the removal of a species from a community can generate different indirect effects depending on the trophic levels that are interacting. Conservation managers eradicating introduced mammals must consider ecosystemwide effects and view island management within a "whole ecosystem" context. It is important to consider unexpected indirect effects from eradications. However, eradication of introduced mammals should not be delayed, especially when direct effects such as predation are a known cause of terminal decline for a threatened species. We use the French Îles Éparses of the Western Indian Ocean, with their various combinations of six introduced mammals, to demonstrate the direct and indirect effects that introduced mammals may have, and how those effects might affect the regionally important seabirds with breeding colonies on the islands. We conclude by making recommendations for the future management of the islands.
