A Simple Predator-Prey Population Model with Rich Dynamics

Dynamics of a delayed predator–prey model with predator migration

Distributed delay is included in a simple predator–prey model in the prey-to-predator biomass conversion term. The delayed term includes a delay-dependent “discount” factor that ensures the predators that do not survive the delay interval, do not contribute to growth of the predator population. A simple model was chosen so that without delay all solutions converge to a globally asymptotically stable equilibrium in order to show the possible effects of delay on the dynamics. If the co-existence equilibrium does not exist, the dynamics of the system is identical to its non-delayed analog. However, with delay, there is a delay-dependent threshold for the existence of the co-existence equilibrium. When the co-existence equilibrium exists, unlike the dynamics of the model without delay, a much wider range of dynamics is possible, including a strange attractor and bi-stability, although the system is uniformly persistent. A bifurcation theory approach is taken, using both the mean delay and the predator death rate as bifurcation parameters. We consider the gamma and the uniform distributions as delay kernels and show that the “discounting” term ensures that the Hopf bifurcations occur in pairs, as was observed in the analogous system with discrete delay (i.e., using the Dirac delta distribution). We show that there are certain features common to all distributions, although the model with different kernels can have a significantly different range of dynamics. In particular, the number of bi-stabilities, the sequence of bifurcations, the criticality of the Hopf bifurcations, and the size of the stability regions can differ. Also, the width of the interval over which the delay history is nonzero seems to have a significant effect on the range of dynamics. Thus, ignoring the delay and/or not choosing the right delay kernel might result in inaccurate modelling predictions.

<p style='text-indent:20px;'>A stage-structured predator-prey model with Crowley-Martin functional response is formulated and analyzed to investigate the impaction of predator maturity delay and predator interference on the dynamics of the system. It is shown that the net reproduction number of the system determines its threshold dynamics: the system is permanent if and only if the net reproduction number is larger than one unit, otherwise the predators go extinct; Moreover, given that the system is permanent, the unique coexistence equilibrium is shown to be globally asymptotically stable provided that the predator's interference is large enough; Numerical simulations are also performed to explore the effects of predator maturity delay and predator's interference on the stability of interior equilibrium. It is indicated that an increase of maturity delay of the predator will enlarge its extinction risk and may also enrich dynamics of the system in the sense of bringing stability switches of the coexistence equilibrium; and that an increase of predator's interference may lead to stability switch of coexistence equilibrium from unstable to stable, implying that a large predator's interference can be stabilizing.</p>

Adaptive behaviour and multiple equilibrium states in a predator–prey model

Limit cycles of a predator–prey model with intratrophic predation

This paper presents the bifurcation behaviors of a modified railway wheelset model to explore its instability mechanisms of hunting motion. Equivalent conicity data measured from China high-speed railway vehicle are used to modify the wheelset model. Firstly, the relationships between longitudinal stiffness, lateral stiffness, equivalent conicity and critical speed are taken into account by calculating the real parts of the eigenvalues of the Jacobian matrix and Hurwitz criterion for the corresponding linear model. Secondly, measured equivalent conicity data are fitted by a nonlinear function of the lateral displacement rather than are considered as a constant as usual. Nonlinear wheel–rail force function is used to describe the wheel–rail contact force. Based on these modifications, a modified railway wheelset model with nonlinear equivalent conicity and wheel–rail force is set up, and then, some instability mechanisms of China high-speed train vehicle are investigated based on Hopf bifurcation, fold (limit point) bifurcation of cycles, cusp bifurcation of cycles, Neimark–Sacker bifurcation of cycles and 1:1 resonance. In particular, fold bifurcation of cycles can produce a vast effect on the hunting motion of the modified wheelset model. One of the main reasons leading to hunting motion is due to the fold bifurcation structure of cycles, in which stable limit cycles and unstable limit cycles may coincide, and multiple nested limit cycles appear on a side of fold bifurcation curve of cycles. Unstable hunting motion mainly depends on the coexistence of equilibria and limit cycles and their positions; if the most outward limit cycle is stable, then the motion of high-speed vehicle should be safe in a reasonable range. Otherwise, if the initial values are chosen near the most outward unstable limit cycle or the system is perturbed by noises, the high-speed vehicle will take place unstable hunting motion and even lead to serious train derailment events. Therefore, in order to control hunting motions, it may be the easiest way in theory to guarantee the coexistence of the inner stable equilibrium and the most outward stable limit cycle in a wheelset system.

The topological structures of complex networks have been playing an important role on the epidemic spreading. There has been several studies of pairwise epidemic models on adaptive networks with Poisson distribution, all of which have shown that the rewiring behaviors can lead to complex dynamics numerically or analytically. However, the triples approximation formula under Poisson distribution overlooked the degree of center node of triples which has dramatic effects on the structures. Therefore in this paper, through a new moment closure incorporating the effect of center node's degree, we study how the topological structures of adaptive networks influences epidemic dynamics. The SIS pairwise epidemic model is first closed by the new triple approximation formula, then we transform the model into an equivalent nondimensionalized three dimensional system. By the qualitative theory and the stability theory of ordinary differential equations, the basic reproduction number R0 of the model is obtained, the existence and stabilities of the equilibria are analyzed. Moreover, we prove that the model exhibits transcritical forward bifurcation, backward bifurcation, saddle-node bifurcation and Hopf bifurcation using the methods of bifurcation theory. In addition, by a numerical example, the normal form of Hopf bifurcation and the first Lyapunov coefficient are derived, which show that a stable limit cycle can bifurcate from the endemic equilibrium with larger epidemicity. Our study show that the adaptive behavior can lead to rich dynamics on epidemic transmission, including oscillation and bistability. Finally the numerical simulations which is consistent with the analytical results above are given.

Dynamical analysis of a model of social behavior: Criminal vs non-criminal population

BackgroundLittle is known about the impact of prey sexual dimorphism on predator-prey dynamics and the impact of sex-selective harvesting and trophy hunting on long-term stability of exploited populations.Methodology and Principal FindingsWe review the quantitative evidence for sex-selective predation and study its long-term consequences using several simple predator-prey models. These models can be also interpreted in terms of feedback between harvesting effort and population size of the harvested species under open-access exploitation. Among the 81 predator-prey pairs found in the literature, male bias in predation is 2.3 times as common as female bias. We show that long-term effects of sex-selective predation depend on the interplay of predation bias and prey mating system. Predation on the ‘less limiting’ prey sex can yield a stable predator-prey equilibrium, while predation on the other sex usually destabilizes the dynamics and promotes population collapses. For prey mating systems that we consider, males are less limiting except for polyandry and polyandrogyny, and male-biased predation alone on such prey can stabilize otherwise unstable dynamics. On the contrary, our results suggest that female-biased predation on polygynous, polygynandrous or monogamous prey requires other stabilizing mechanisms to persist.Conclusions and SignificanceOur modelling results suggest that the observed skew towards male-biased predation might reflect, in addition to sexual selection, the evolutionary history of predator-prey interactions. More focus on these phenomena can yield additional and interesting insights as to which mechanisms maintain the persistence of predator-prey pairs over ecological and evolutionary timescales. Our results can also have implications for long-term sustainability of harvesting and trophy hunting of sexually dimorphic species.

In this paper, we investigate a predator–prey model with Holling-II functional response, Allee effect and constant-yield predator harvesting, by comparing the differences between [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the harvesting rate of predators. For system without harvesting, the Allee effect leads to population extinction. The system has at most one positive equilibrium and has a supercritical Hopf bifurcation which depends on the natural mortality rate of predators. Besides, by using normal form theory, we show that the system with [Formula: see text] reveals rich dynamic properties, including saddle–node bifurcation, Hopf bifurcation and Bogdanov–Takens bifurcation, where numerical simulations are presented to demonstrate the Bogdanov–Takens bifurcation of codimension 2 with a limit cycle and a homoclinic cycle. The system can generate up to two positive equilibria with the changes of [Formula: see text], which indicates that appropriate predator harvesting can assist in regulating the ecosystem. We then give the optimal harvesting strategy by using Pontryagin’s maximum principle. Finally, numerical simulations are performed to validate the functions of Allee effect and harvesting. Theoretical studies and numerical simulations demonstrate that the Allee effect can lead to species extinction and highlight the role of appropriate harvesting in controlling the stability of the system.

A discrete delay is included to model the time between the capture of the prey and its conversion to viable biomass in the simplest classical Gause type predator-prey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of period doublings, eventually leading to chaos. The resulting periodic orbits and the strange attractor resemble their counterparts for the Mackey-Glass equation. Due to the global stability of the system without delay, this compli- cated dynamics can be solely attributed to the introduction of the delay. Since many models include predator-prey like interactions as submodels, this study emphasizes the importance of understanding the implications of overlooking delay in such models on the reliability of the model-based predictions, espe- cially since temperature is known to have an effect on the length of certain delays.

In this paper, we propose a stage-structured predator-prey model with migrations among patches in an n-patch environment. The net reproduction number for each patch in isolation is obtained along with the net reproduction number of the system of patches, ℛ0. Inequalities describing the relationship among these numbers are also given. Furthermore, threshold dynamics determined by ℛ0 is established: the predator dies out if ℛ0<1 while the predator persists if ℛ0>1. Focusing on the case with two patches, we obtain that the dispersal decreases the net reproduction number ℛ0. By numerical simulations, we find that the dispersal may be a good thing or a bad thing because the dispersal could make the predator population thrive or extinct, and hence we might seek steady state in the ecological environment by controlling parameters related to the prey and the predator.

Related studies indicate that the Rosenzweig–MacArthur predator–prey model with constant search rate exhibits the paradox of enrichment, while variable search rate may avoid this event. We investigate the effect of variable search rate on the dynamics of the predator–prey model, a complete and explicit classification of global dynamics is characterized using carrying capacity [Formula: see text] and half-saturation constant [Formula: see text] as parameters. First, this is a trichotomy result, the boundary equilibrium or the positive equilibrium is globally asymptotically stable; otherwise, there exists a globally stable limit cycle. Furthermore, we demonstrate that for the case where the limit cycle is stable, the predator can adjust search rate to restore the positive equilibrium to stable state, thereby avoiding the occurrence of the paradox of enrichment. Specifically, we adopt a new planar analysis method that differs from classical ways, proving that local stability of the positive equilibrium implies its global stability. Finally, numerical simulation is conducted to verify the results of theoretical analysis. Our results can help understand the evolutionary mechanisms through which organisms adapt to environmental changes in some sense.

To explore the idea that the dynamics of the zooplankter Daphnia and its algal food supply can be accounted for by simple predator-prey models, we tested the ability of such models to predict how the average densities change along a gradient of nutrient enrichment. Total algal carrying capacity is known to increase from nutrient-poor to nutrient-rich lakes. Here we show that this is true also for edible algae alone. We also provide evidence that the realized per capita growth rate of edible algae increases with trophy (trend 1). A synthesis of laboratory and field data suggests two other trends associated with increasing nutrient status: (2) Daphnia's attack rate decreases in response to increasing concentrations of inedible algae; and (3) its death rate increases in response to greater predation pressure. In models of the Lotka-Volterra type, either trend 2 or a combination of trends 1 and 3 leads to the prediction that the abundance of both Daphnia and their algal prey should increase along a nutrient gradient in natural habitats where inedible algae and predators of Daphnia both occur. In contrast, where these features are missing, the original "paradox of enrichment" prediction is expected: Daphnia density should increase but algal density should not. Our analysis of more than 30 study-years of data from natural lakes confirmed the first of these predictions; results from outdoor artificial habitats, lacking inedible algae and predators of Daphnia, confirmed the second. We cannot evaluate the relative importance of the two mechanisms competing to explain the result from natural habitats, but our analyses suggest new observations and experiments that could test these mechanisms. These results are consistent with our general hypothesis that the dynamics of the Daphnia-algae interaction can be explained by simple predator-prey models. In particular, they support two assumptions: (1) other species in the system can be modeled simply as part of the environment of the predator-prey interaction; and (2) variation in the dynamics from one habitat to another results from merely quantitative differences in parameter values and does not require us to postulate qualitative, or "structural," changes among habitats.

A recent increase in jellyfish populations: A predator-prey model and its implications