Analysis of a competitive prey–predator system with a prey refuge

Gauss's competitive exclusive principle states that two competing species having analogous environment cannot usually occupy the same space at a time but in order to exploit their common environment in a different manner, they can co-exist only when they are active in different times. On the other hand, several studies on predators in various natural and laboratory situations have shown that competitive coexistence can result from predation in a way by resisting any one prey species from becoming sufficiently abundant to outcompete other species such that the predator makes the coexistence possible. It has also been shown that the use of refuges by a fraction of the prey population exerts a stabilizing effect in the interacting population dynamics. Further, the field surveys in the Sundarban mangrove ecosystem reveal that two detritivorous fishes, viz. Liza parsia and Liza tade (prey population) coexist in nature with the presence of the predator fish population, viz. Lates calcarifer by using refuges.     

Viable populations in a prey–predator system

 Lotka–Volterra equations are considered a dynamical game, where the phenotypes of the predator and of the prey can vary. This differs from the usual procedure of specifying as a priori laws according to which strategies are supposed to change. The question at stake is the survival of each of the species, instead of the maximization of a given pay-off by each player, as it is commonly discussed in games. The predator needs the prey, while the prey can survive without the predator. These obvious and simplistic constraints are enough to shape the regulation of the system: notably, the largest closed set of initial conditions can be delineated, from which there exists at least one evolutionary path where the population can avoid extinction forever. To these so-called viable trajectories, viable strategies are associated, respectively for the prey or for the predator. A coexistence set can then be defined. Within this set and outside the boundary, strategies can vary arbitrarily within given bounds while remaining viable, whereas on the boundary, only specific strategies can guarantee the viability of the system. Thus, the largest set can be determined, outside of which strategies will never be flexible enough to avoid extinction. Received 2 May 1995; received in revised form 15 August 1995     

Effects of the probability of a predator catching prey on predator–prey system stability

To understand the effect of the probability of a predator catching prey, P_catch, on the stability of the predator–prey system, a spatially explicit lattice model consisting of predators, prey, and grass was constructed. The predators and prey randomly move on the lattice space, and the grass grows according to its growth probability. When a predator encounters prey, the predator eats the prey in accordance with the probability P_catch. When a prey encounters grass, the prey eats the grass. The predator and prey give birth to offspring according to a birth probability after eating prey or grass, respectively. When a predator or prey is initially introduced or newly born, its health state is set at a high given value. This health state decreases by one with every time step. When the state of an animal decreases to less than zero, the individual dies and is removed from the system. Population densities for predator and prey fluctuated significantly according to P_catch. System stability was characterized by the standard deviation ? of the fluctuation. The simulation results showed that ? for predators increased with an increase of P_catch; ? for prey reached a maximum at P_catch = 0.4; and ? for grass fluctuated little regardless of P_catch. These results were due to the tradeoff between P_catch and the predator–prey encounter rate, which represents the degree of interaction between predator and prey and the average population density, respectively.     

The complex dynamics of a diffusive prey–predator model with an Allee effect in prey

This paper investigates complex dynamics of a predator–prey interaction model that incorporates: (a) an Allee effect in prey; (b) the Michaelis–Menten type functional response between prey and predator; and (c) diffusion in both prey and predator. We provide rigorous mathematical results of the proposed model including: (1) the stability of non-negative constant steady states; (2) sufficient conditions that lead to Hopf/Turing bifurcations; (3) a prior estimates of positive steady states; (4) the non-existence and existence of non-constant positive steady states when the model is under zero-flux boundary condition. We also perform completed analysis of the corresponding ODE model to obtain a better understanding on effects of diffusion on the stability. Our analytical results show that the small values of the ratio of the prey's diffusion rate to the predator's diffusion rate are more likely to destabilize the system, thus generate Hopf-bifurcation and Turing instability that can lead to different spatial patterns. Through numerical simulations, we observe that our model, with or without Allee effect, can exhibit extremely rich pattern formations that include but not limit to strips, spotted patterns, symmetric patterns. In addition, the strength of Allee effects also plays an important role in generating distinct spatial patterns.     

Parametric Analysis of a Predator–prey System Stabilized by a Top Predator

We present a complete parametric analysis of a predator–prey system influenced by a top predator. We study ecosystems with abundant nutrient supply for the prey where the prey multiplication can be considered as proportional to its density. The main questions we examine are the following: (1) Can the top predator stabilize such a system at low densities of prey? (2) What possible dynamic behaviors can occur? (3) Under which conditions can the top predation result in the system stabilization? We use a system of two nonlinear ordinary differential equations with the density of the top predator as a parameter. The model is investigated with methods of qualitative theory of ODEs and the theory of bifurcations. The existence of 12 qualitatively different types of dynamics and complex structure of the parametric space are demonstrated. Our studies of phase portraits and parametric diagrams show that a top predator can be an important factor leading to stabilization of the predator-prey system with abundant nutrient supply. Although the model here is applied to the plankton communities with fish (or carnivorous zooplankton) as the top trophic level, the general form of the equations allows applications of our results to other ecological systems.     

Cooperative hunting in a discrete predator–prey system II

ABSTRACT

We investigate a discrete-time predator–prey system with cooperative hunting in the predators proposed by Chow et al. by determining local stability of the interior steady states analytically in certain parameter regimes. The system can have either zero, one or two interior steady states. We provide criteria for the stability of interior steady states when the system has either one or two interior steady states. Numerical examples are presented to confirm our analytical findings. It is concluded that cooperative hunting of the predators can promote predator persistence but may also drive the predator to a sudden extinction.     

Evolution towards oscillation or stability in a predator–prey system

We studied a prey–predator system in which both species evolve. We discuss here the conditions that result in coevolution towards a stable equilibrium or towards oscillations. First, we show that a stable equilibrium or population oscillations with small amplitude is likely to occur if the prey''s (host''s) defence is effective when compared with the predator''s (parasite''s) attacking ability at equilibrium, whereas large-amplitude oscillations are likely if the predator''s (parasite''s) attacking ability exceeds the prey''s (host''s) defensive ability. Second, a stable equilibrium is more likely if the prey''s defensive trait evolves faster than the predator''s attack trait, whereas population oscillations are likely if the predator''s trait evolves faster than that of the prey. Third, when the adaptation rates of both species are similar, the amplitude of the fluctuations in their abundances is small when the adaptation rate is either very slow or very fast, but at an intermediate rate of adaptation the fluctuations have a large amplitude. We also show the case in which the prey''s abundance and trait fluctuate greatly, while those of the predator remain almost unchanged. Our results predict that populations and traits in host–parasite systems are more likely than those in prey–predator systems to show large-amplitude oscillations.     

Mechanosensory‐based orientation to elevated prey by a benthic fish

Lake Michigan mottled sculpin, Cottus bairdi, respond to both live and artificial (e.g., vibrating sphere) prey with an unconditioned movement towards the source of vibration, followed by a step‐by‐step approach and final strike at the source. In addition to these well‐studied, whole‐body movements along the horizontal plane of the substrate, sculpin exhibit a little‐studied behavior in which the vertical position of the fish's head can vary from being flush with the substrate to several cm's above the substrate. To test the hypothesis that sculpin can determine source elevation via mechanosensory cues, we measured head elevation of blinded fish as a function of source elevation and distance as fish approached a small (3 mm radius), 50 Hz vibrating sphere. At distances associated with pre‐strike positions (< 2 cm), head elevations were positively correlated with source elevation before but not after pharmacological blocking of the lateral line with CoCl₂. These results demonstrate that sculpin are able to determine source elevation using mechanosensory cues alone and that in the absence of visual and olfactory cues, vertical orientation of the head towards the source requires the lateral line system.     

Sensory-based niche partitioning in a multiple predator–multiple prey community

Many predators and parasites eavesdrop on the communication signals of their prey. Eavesdropping is typically studied as dyadic predator–prey species interactions; yet in nature, most predators target multiple prey species and most prey must evade multiple predator species. The impact of predator communities on prey signal evolution is not well understood. Predators could converge in their preferences for conspicuous signal properties, generating competition among predators and natural selection on particular prey signal features. Alternatively, predator species could vary in their preferences for prey signal properties, resulting in sensory-based niche partitioning of prey resources. In the Neotropics, many substrate-gleaning bats use the mate-attraction songs of male katydids to locate them as prey. We studied mechanisms of niche partitioning in four substrate-gleaning bat species and found they are similar in morphology, echolocation signal design and prey-handling ability, but each species preferred different acoustic features of male song in 12 sympatric katydid species. This divergence in predator preference probably contributes to the coexistence of many substrate-gleaning bat species in the Neotropics, and the substantial diversity in the mate-attraction signals of katydids. Our results provide insight into how multiple eavesdropping predator species might influence prey signal evolution through sensory-based niche partitioning.     

Factors promoting polygyny in European birds of prey—a hypothesis

Summary Polygyny is known in at least nine (out of 36) European raptor (Accipitriformes and Falconiformes) and seven (out of 13) owl (Strigiformes) species that hunt mobile prey. The hypothesis put forward here suggests that abundant food supply and nomadic tactics of breeding dispersal are crucial factors promoting polygyny in birds of prey. The hypothesis predicts that: (1) polygyny is more common in rodent-eating birds of prey than in bird-eating ones; (2) polygyny is more frequent in good vole years than in poor ones; (3) the frequency of polygyny in vole-eating species should increase northwards in Europe, as the densities of voles in the peak phase increase in that direction; (4) the frequency of polygyny and harem size should be increased by supplementary feeding; and (5) polygyny is more common in nomadic birds of prey with annual pair bonds and weak territoriality than in resident birds of prey with longerterm pair bonds and stronger territoriality. A majority of the available data is consistent with predictions 1–3 and 5, but data on prediction 4 are scanty. Further studies on ringed birds of prey are needed to test the validity of the hypothesis.     

Predator–prey system with strong Allee effect in prey

Global bifurcation analysis of a class of general predator–prey models with a strong Allee effect in prey population is given in details. We show the existence of a point-to-point heteroclinic orbit loop, consider the Hopf bifurcation, and prove the existence/uniqueness and the nonexistence of limit cycle for appropriate range of parameters. For a unique parameter value, a threshold curve separates the overexploitation and coexistence (successful invasion of predator) regions of initial conditions. Our rigorous results justify some recent ecological observations, and practical ecological examples are used to demonstrate our theoretical work.     

Does prey preference change as a result of prey species being presented together? Analysis of prey selection by the predatory mite Typhlodromus pyri (Acarina: Phytoseiidae)

Summary Prey-selection behaviour of the phytoseiid mite Typhlodromus pyri Scheuten was analysed with a Markovtype model of feeding-state dynamics and feeding-state dependent searching behaviour (Sabelis 1981, 1986, 1989; Metz and Van Batenburg 1985a, b). All behavioural characteristics of the predator which are independent of the feeding state were represented by one parameter. The remaining feeding-state dependent characteristics were represented by a function of the feeding state, with one parameter. The best parameter values to describe a predator-prey interaction were determined by fitting the model to the predation rates in monocultures. Under the assumption that the parameter values are not dependent on the composition of prey species supply, the diet of the predators in mixed cultures was predicted from parameters estimated in monoculture experiments.Two prey types, apple rust mite (Aculus schlechtendali (Nalepa)) adults and European red spider mite (Panonychus ulmi (Koch)) larvae were studied. A large discrepancy was observed between calculated and experimentally determined predation rates of T. pyri in mixed cultures: the predators actually killed 3–7 times more P. ulmi larvae than was predicted by the model.The large difference between observed and predicted predation rates in mixed cultures cannot be explained by changes in the behaviour of the prey species as a result of being together. Therefore, it seems likely that the prey selection behaviour of the predator was different when prey species were presented together than when presented singly. Apparently the predatory mite T. pyri prefers P. ulmi to S. schlechtendali.     

Effect of prey type and inorganic turbidity on littoral predator–prey interactions in a shallow lake: an experimental approach

Predation often represents the prevailing process shaping aquatic ecosystems. As foraging and antipredatory behaviour frequently relate to vision, turbidity may often impair the interactions between the predator and its prey, depending on prey type and source and level of turbidity. We studied the effect of inorganic turbidity (0–30 NTU) on the effectiveness of fish feeding on two types of prey in different habitats: free-swimming cladoceran (Daphnia pulex) in open water and plant-associated cladoceran (Sida crystallina) attached to Nuphar lutea leaves. For the planktivore, we used vision-oriented perch (Perca fluviatilis) common in the littoral zone of temperate lakes. In our study, increasing inorganic turbidity did not appear to initiate any significant change in the feeding efficiency of perch on free-swimming Daphnia pulex. However, we saw a markedly different feeding efficiency when perch targeted plant-attached Sida crystallina. Our results substantiate that floating-leaved macrophytes in turbid lakes may provide a favourable habitat for plant-attached cladocerans.     

Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

Mechanisms and scenarios of pattern formation in predator–prey systems have been a focus of many studies recently as they are thought to mimic the processes of ecological patterning in real-world ecosystems. Considerable work has been done with regards to both Turing and non-Turing patterns where the latter often appears to be chaotic. In particular, spatiotemporal chaos remains a controversial issue as it can have important implications for population dynamics. Most of the results, however, were obtained in terms of ‘traditional’ predator–prey models where the per capita predation rate depends on the prey density only. A relatively new family of ratio-dependent predator–prey models remains less studied and still poorly understood, especially when space is taken into account explicitly, in spite of their apparent ecological relevance. In this paper, we consider spatiotemporal pattern formation in a ratio-dependent predator–prey system. We show that the system can develop patterns both inside and outside of the Turing parameter domain. Contrary to widespread opinion, we show that the interaction between two different type of instability, such as the Turing–Hopf bifurcation, does not necessarily lead to the onset of chaos; on the contrary, the emerging patterns remain stationary and almost regular. Spatiotemporal chaos can only be observed for parameters well inside the Turing–Hopf domain. We then investigate the relative importance of these two instability types on the onset of chaos and show that, in a ratio-dependent predator–prey system, the Hopf bifurcation is indeed essential for the onset of chaos whilst the Turing instability is not.     

Undamped oscillations in prey–predator models on a finite size lattice

Sustained oscillation is frequently observed in population dynamics of biospecies. The oscillation comes not only from deterministic but also from stochastic characteristics. In the present article, we deal with a finite size lattice which contains prey and predator. The interaction between a pair of lattice points is carried out by two different methods; local and global interactions. In the former, interaction occurs between adjacent sites, while in the latter interaction takes place between any pair of lattice sites. It is found that both systems exhibit undamped oscillations. The amplitude of oscillation decreases with the increase of the total lattice sites. In the case of global interaction, we can present a stochastic differential equation which is composed of two factors, i.e., the Lotka–Volterra equation with density dependence and noise term. The quantitative agreement between theory and simulation results of global interaction is almost perfect. The stochastic theory qualitatively expresses characteristics of sustainable oscillation for local interaction.     

Effects of rapid prey evolution on predator–prey cycles

We study the qualitative properties of population cycles in a predator-prey system where genetic variability allows contemporary rapid evolution of the prey. Previous numerical studies have found that prey evolution in response to changing predation risk can have major quantitative and qualitative effects on predator-prey cycles, including: (1) large increases in cycle period, (2) changes in phase relations (so that predator and prey are cycling exactly out of phase, rather than the classical quarter-period phase lag), and (3) "cryptic" cycles in which total prey density remains nearly constant while predator density and prey traits cycle. Here we focus on a chemostat model motivated by our experimental system (Fussmann et al. in Science 290:1358-1360, 2000; Yoshida et al. in Proc roy Soc Lond B 424:303-306, 2003) with algae (prey) and rotifers (predators), in which the prey exhibit rapid evolution in their level of defense against predation. We show that the effects of rapid prey evolution are robust and general, and furthermore that they occur in a specific but biologically relevant region of parameter space: when traits that greatly reduce predation risk are relatively cheap (in terms of reductions in other fitness components), when there is coexistence between the two prey types and the predator, and when the interaction between predators and undefended prey alone would produce cycles. Because defense has been shown to be inexpensive, even cost-free, in a number of systems (Andersson et al. in Curr Opin Microbiol 2:489-493, 1999: Gagneux et al. in Science 312:1944-1946, 2006; Yoshida et al. in Proc Roy Soc Lond B 271:1947-1953, 2004), our discoveries may well be reproduced in other model systems, and in nature. Finally, some of our key results are extended to a general model in which functional forms for the predation rate and prey birth rate are not specified.     

Ontogenetic shifts in a prey’s chemical defences influence feeding responses of a snake predator

Foraging theory suggests that predator responses to potential prey should be influenced by prey chemical defences, but the effects of ontogenetic variation in such defences on prey vulnerability to predators remain unclear. Cane toads (Rhinella marina) are toxic to anurophagous snakes, including the keelback (Tropidonophis mairii, a natricine colubrid that occurs within the toads' invasive range in Australia). Toxin levels and diversity change through toad ontogeny, decreasing from the egg stage to metamorphosis, then increasing in postmetamorphic toads. If the toxin content of a prey item influences predator responses, we predict that keelbacks should exhibit selective predation on toads close to metamorphosis. The results of our laboratory trials on adult (field-collected, and thus toad-experienced) and hatchling (laboratory-incubated, and thus toad-naive) keelbacks supported this prediction. The snakes selectively consumed later-stage rather than earlier-stage tadpoles, and earlier-stage rather than later-stage metamorphs. Our data are thus consistent with the hypothesis that ontogenetic changes in toxin content can affect individuals' vulnerability to predation.     

Spatiotemporal patterns provoked by environmental variability in a predator–prey model

The emergence of spatiotemporal patterns in the distribution of species is one of the most striking phenomena in ecology and nonlinear science. Since it is known that spatial inhomogeneities can significantly affect the dynamics of ecological populations, in the present paper we investigate the impact of environmental variability on the formation of patterns in a spatially extended predator–prey model. In particular, we utilize a predator–prey system with a Holling III functional response and introduce random spatial variations of the kinetic parameter signifying the intrinsic growth rate of the prey, reflecting the impact of a heterogeneous environment. Our results reveal that in the proximity of the Hopf bifurcation environmental variability is able to provoke pattern formation, whereby the coherence of the patterns exhibits a resonance-like dependence on the variability strength. Furthermore, we show that the phenomenon can only be observed if the spatial heterogeneities exhibit large enough regions with high growth rates of the prey. Our findings thus indicate that variability could be an essential pattern formation mechanism of the populations.     

Predator–prey ratios on cocoa along a land-use gradient in Indonesia

Tropical landscapes are dominated by agroecosystems, but the potential value of agroecosystems for the survival of species is often overlooked. In agroecosystems, species conservation is especially important when functional groups such as predators are affected. In Central Sulawesi, we sampled arthropods on cocoa in a gradient of land-use intensity from extensively used forest gardens to intensively used agroforestry systems. The abundance and diversity of all arthropods did not correlate with land-use intensity, so human impact was not followed by high species losses. However, the number of species and abundance of the phytophagous arthropods increased and that of the entomophagous arthropods decreased with land-use intensity. The reduced predator–prey ratio in intensified systems can be related to their reduced species richness of shade trees and the changed microclimate (increased temperature, decreased humidity and canopy cover). In conclusion, transformation of traditional into intensified agroforestry systems had a great impact on arthropod community structure on cocoa. Since predator–prey ratios decreased with increasing land-use intensity, local farmers should have least pest problems in the traditionally diversified agroforestry systems.     

Assessing prey competition in fossil carnivore communities — a scenario for prey competition and its evolutionary consequences for tigers in Pleistocene Java

We assess prey competition between various carnivore species that overlapped in time and geographic range during the Early to Late Pleistocene of Java, using a three step procedure. Our sample encompasses 5 carnivore species–two pantherines (Panthera tigris, P. pardus); a single hyaenid (Hyaena brevirostris); and two canid species (Cuon alpinus, Cuon (Mececyon) trinilensis)–each of which occurs in one or more faunal levels (Ci Saat, Trinil HK, Kedung Brubus, Ngandong, and Punung) in the well-documented Stegodon–Homo erectus Fauna. The tiger is the only carnivore that occurs in all faunal levels studied here. Since changing body mass can lessen competition, monitoring tiger body mass through the chronological sequence–as other species enter or leave the fossil record–provides an opportunity to assess the effects of competition. First, we estimate body mass of each individual in these species using regression equations based on measurement of skeletal elements. Second step, from these estimates of individual body mass we derive prey focus masses (PFM) and prey mass spectra (PMS). Third, we examine pairwise PMS overlap of coexisting carnivore species, deriving a quantitative Competition Index (CI) that expresses the degree to which a specific carnivore is affected by the presence of another carnivore species. The CI provides a measure of the consequences of changes in the carnivore communities upon each species and also allows us to evaluate the impact of different hunting strategies on competition. We suggest the significant increase in body mass of tigers in the Ngandong faunal level reflects unusually intense competition among carnivores in the preceding Kedung Brubus level.