Two-species Predator-prey Model Research Articles

Analysis of a stochastic two-species predator-prey system in two-patch environments with Ornstein-Uhlenbeck process

To characterize the effects of diffusion and environmental noise on population dynamics, in the paper, we first develop a stochastic two-species predator-prey model in two-patch environments, where the fluctuations in the environment are depicted by an Ornstein-Uhlenbeck process. Then we analyze the dynamical behavior of the stochastic model in detail, including the existence and uniqueness of the global solution, the pth moment boundedness, asymptotic pathwise estimation and the existence of a stationary distribution. It is worth noting that by solving the six-dimensional algebraic equations corresponding to the stochastic model, we get the approximate expression of the probability density function around the quasi-coexistence equilibrium of the stochastic system. Finally, a series of numerical simulations are presented to confirm the feasibility and correctness of our theoretical analysis results. Our findings show the importance of considering the effect of this type of stochastic process on the population dynamics.

Population dynamics in a reaction–diffusion-advection predator–prey model with Beddington–DeAngelis functional response

In this paper, we consider a two-species predator–prey model in advective heterogeneous environments with the Beddington–DeAngelis interaction term, where the Danckwerts boundary conditions are imposed. Applying the comparison principle for predator–prey system and persistence theory, we draw a clear picture on the long-time dynamics, which also indicates the existence and non-existence of the coexistence steady state solutions. Specially, the uniqueness of the coexistence steady state solutions is considered under certain conditions.

Rich dynamics of a discrete two dimensional predator–prey model using the NSFD scheme

In this paper, we consider a two-species predator–prey model with Holling type III functional response and non-linear predator harvesting. The proposed model is discretized using a non-standard finite difference scheme (NSFD). The stability of different equilibrium points are analyzed. Also, the conditions of various types of bifurcations likely: Transcritical, Neimark–Sacker bifurcation (NSB), and Flip (Period doubling) bifurcation (PDB) have been established along with chaos control strategies. The numerical results indicate that the system exhibits different patterns of solutions, including single, two, and higher periodicity. Using Lyapunov exponents and bifurcation diagrams, chaotic solutions are verified. Two model parameters were drawn simultaneously in the attractor basin, which yielded different periodic solutions compared to the continuous dynamical system. Lastly, the pole placement method (PPM) has been used to control chaos in the proposed discrete ecological model.

Fear-driven extinction and (de)stabilization in a predator-prey model incorporating prey herd behavior and mutual interference

<abstract><p>The indirect effect of predation due to fear has proven to have adverse effects on the reproductive rate of the prey population. Here, we present a deterministic two-species predator-prey model with prey herd behavior, mutual interference, and the effect of fear. We give conditions for the existence of some local and global bifurcations at the coexistence equilibrium. We also show that fear can induce extinction of the prey population from a coexistence zone in finite time. Our numerical simulations reveal that varying the strength of fear of predators with suitable choice of parameters can stabilize and destabilize the coexistence equilibrium solutions of the model. Further, we discuss the outcome of introducing a constant harvesting effort to the predator population in terms of changing the dynamics of the system, in particular, from finite time extinction to stable coexistence.</p></abstract>

Stabilization for a class of delay systems via Z-type control

This paper investigates the Z-type control method to stabilize species populations and make them converge to desired values in a two-species predator–prey model with time delay in the prey species’ growth rate. Direct controllers that assume both species’ growth rates can be altered through human intervention and indirect controllers that assume only one species’ growth can be changed at will are constructed. The Z-type method of construction is theoretically justified, and the potential for certain non-linear activation functions to speed up the control process is highlighted. A general procedure for discretizing state-space systems using the Euler forward formula is described. The discrete versions of the directly and indirectly controlled systems are provided. All Z-type controllers constructed are validated through numerical simulations.

Predator-prey systems in open advective heterogeneous environments with Holling-Tanner interaction term

In this paper, we consider a two-species predator-prey model in advective heterogeneous environments with the Holling-Tanner interaction term, where the Danckwerts boundary conditions are imposed. Existence, uniqueness, and multiplicity of positive steady state solutions are investigated by using the degree theory and super-sub solution method for predator-prey systems and analyzing the limiting profile. Using the comparison principle for parabolic equations, persistence theory, and constructing Lyapunov functions, we establish various results on the long-time dynamics.

Impact of adult predator incited fear in a stage-structured prey–predator model

A two-species predator–prey model where the predators are separated into juvenile and mature predators has been considered in this article. Here, functional response in linear form is considered to know the impact of adult predator-induced fear in the stage-structured prey–predator model. Positivity and boundedness of the model system have been checked. Persistence conditions and existence condition(s) of each equilibria have been derived. The local and global stability analysis have been implemented both analytically and numerically along with Hopf bifurcation analysis with direction. From the analysis of the model system, it is observed that the mature predator-induced fear and rate of transition from juvenile (minor) predator to mature predator plays a crucial role in controlling the system dynamics.

A modeling study of predator–prey interaction propounding honest signals and cues

Honest signals and cues have been observed as part of interspecific and intraspecific communication among animals. Recent theories suggest that existing signaling systems have evolved through natural selection imposed by predators. Honest signaling in the interspecific communication can provide insight into the evolution of anti-predation techniques. In this work, we introduce a deterministic three-stage, two-species predator–prey model, which modulates the impact of honest signals and cues on the interacting populations. The model is built from a set of first principles originated from signaling and social learning theory in which the response of predators to transmitted honest signals or cues is determined. The predators then use the signals to decide whether to pursue the attack or save their energy for an easier catch. Other members from the prey population that are not familiar with signaling their fitness observe and learn the technique. Our numerical bifurcation analysis indicates that increasing the predator’s search rate and the corresponding assimilation efficiency gives a journey from predator–prey abundance and scarcity, a stable transient cycle between persistence and near-extinction, a homoclinic orbit pointing towards extinction, and ultimately, a quasi-periodic orbit. A similar discovery is met under the increment of the prey’s intrinsic birth rate and carrying capacity. When both parameters are of sufficiently large magnitudes, the separator between honest signal and cue takes the similar journey from a stable equilibrium to a quasi-periodic orbit as it increases. In the context of modeling, we conclude that under prey abundance, transmitting error-free honest signals leads to not only a stable but also more predictable predator–prey dynamics.

Spatial Scales of Population Synchrony in Predator-Prey Systems

Many species show synchronous fluctuations in population size over large geographical areas, which are likely to increase their regional extinction risk. Here we examine how the degree of spatial synchrony in population dynamics is affected by trophic interactions using a two-species predator-prey model with spatially correlated environmental noise. We show that the predator has a larger spatial scale of population synchrony than the prey if the population fluctuations of both species are mainly determined by the direct effect of stochastic environmental variations in the prey. This result implies that in ecosystems regulated from the bottom up, the spatial scale of synchrony of the predator population increases beyond the scale of the spatial autocorrelation in the environmental noise and in the prey fluctuations. Harvesting the prey increases the spatial scale of population synchrony of the predator, while harvesting the predator reduces the spatial scale of the population fluctuations of its prey. Hence, the development of sustainable harvesting strategies should also consider the impact on unharvested species at other trophic levels as well as human perturbations of ecosystems, whether the result of exploitation or an effect on dispersal processes, as they can affect food web structures and trophic interactions over large geographical areas.

Bifurcation analysis of coexistent state in a delayed two-species predator-prey model

ABSTRACTIn this paper, we consider a delayed two-species predator-prey model with general functional response under the homogeneous Neumann boundary condition. We discuss the stability of the trivial and semi-trivial solutions and obtain the spatially nonhomogeneous bifurcation solutions stemming from the semi-trivial trivial solutions and . Besides, the stability and some results of Hopf bifurcation at the spatially nonhomogeneous bifurcation steady-state solutions are investigated by analyzing the distribution of the eigenvalues. The method we applied here is mainly based on spectral analysis, comparison principle, Lyapunov-Schmidt reduction, and bifurcation theory.

Do prey handling predators really matter: Subtle effects of a Crowley-Martin functional response

Predator interference, or a decline in the per predator consumption rate as predator density increases, is generally considered a stabilizing mechanism in two-species predator-prey models. There is significant debate, as to whether prey handling predators, might interfere in the hunting process of prey searching predators, or whether these are mutually exclusive events. In the current manuscript, a three species food chain model, with strong top predator interference is considered. We prove that in terms of explosive instability/finite time blow up, sufficient interference by prey handling predators always tends to destabilize the system. The dynamics of a time delayed version, as well as the spatially explicit model are also explored. We use our results to comment on a certain paradox in ecological theory, as well as provide further insight into the nature of predator interference, and exploding populations of invasive species.

Logistic equation is a simple stochastic carrying capacity

The logistic model has long been used in ecological modelling for its simplicity and effectiveness. Variations on the logistic model are prolific but, to date, there are a limited number of models that incorporate the stochastic nature of the carrying capacity. This study proposes a modification to the logistic model to incorporate a second differential equation which describes the changes in the carrying capacity, thus treating the carrying capacity as a state variable. The carrying capacity is modelled via a stochastic differential equation that accounts for stochastic ('noisy') variations in the finite resources that the population relies on. The extinction probability distribution, expected solution paths, variance of the solution paths, and distribution of the population are computed using the Monte Carlo method. References R. B. Banks. Growth and Diffusion Phenomena . Springer, 1994. doi:10.1007/978-3-662-03052-3 F. Brauer and C. Castillo-Chavez. Mathematical Models in Population Biology and Epidemiology . Springer, 2001. doi:10.1007/978-1-4614-1686-9 A. Tsoularis and J. Wallace. Analysis of logistic growth models. Math. Biosci. , 179:21–55, 2002. doi:10.1016/S0025-5564(02)00096-2 S. Oppel, G. Hilton, N. Ratcliffe, C. Fenton, J. Daley, G. Gray, J. Vickery and D. Gibbons. Assessing population viability while accounting for demographic and environmental uncertainty. Ecology , 95:1809–1818, 2014. doi:10.1890/13-0733.1 J. E. Cohen. Population growth and earth's human carrying capacity. Science , 269:341–346 1995. doi:10.1126/science.7618100 J. M. Cushing. Periodic time-dependent predator-prey systems. SIAM J. Appl. Math. , 32:82–95 1977. doi:10.1137/0132006 B. D. Coleman. Nonautonomous logistic equation as models of the adjustment of populations to environmental change. Math. Biosci. , 45:159–173, 1979. doi:10.1016/0025-5564(79)90057-9 J. Vandermeer. Seasonal ioschronic forcing of Lotka Volterra equations. Prog. Theor. Phys. , 96:13–28, 1996. doi:10.1143/PTP.96.13 H. M. Safuan, Z. Jovanoski, I. N. Towers and H. S. Sidhu. Exact solution of a non-autonomous logistic population model. Ecol. Model. , 251:99–102, 2013. doi:10.1016/j.ecolmodel.2012.12.016 P. Del Monte-Luna, B. W. Brook, M. J. Zetina-Rejon and V. H. Cruz-Escalona. The carrying capacity of ecosystems. Global Ecol. Biogeogr. , 13:485–495, 2004. doi:10.1111/j.1466-822X.2004.00131.x H. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu. A simple model for the total microbial biomass under occlusion of healthy human skin. 19th International Congress on Modelling and Simulation , Perth, Australia, December 2011, 733–739. http://mssanz.org.au/modsim2011/AA/safuan.pdf H. M. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu. Coupled logistic carrying capacity model. EMAC2011, ANZIAM J. , 53:C172–C184 2012. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/4972 H. M. Safuan, H. S. Sidhu, Z. Jovanoski and I. N. Towers. Impacts of biotic resource enrichment on predator-prey population. B. Math. Biol. , 75:1798–1812, 2013. doi:10.1007/s11538-013-9869-7 H. M. Safuan, H. S. Sidhu, Z. Jovanoski and I. N. Towers. A two-species predator-prey model in an environment enriched by a biotic resource. CTAC2012, ANZIAM J. , 54:C768–C787, 2014 . http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6376 S. M. Henson and J. M. Cushing. The effect of periodic habitat fluctuations on a nonlinear insect population model. J. Math. Biol. , 36:201–226, 1997. doi:10.1007/s002850050098 V. Mendez, I. Llopis, D. Campos and W. Horsthemke. Extinction conditions for isolated populations affected by environmental stochasticity. Theor. Popul. Biol. , 77:250–256, 2010. doi:10.1016/j.tpb.2010.02.006 P. Foley. Predicting extinction times from environmental stochasticity and carrying capacity. Conserv. Biol. , 8:124–137, 1994. http://www.jstor.org/stable/2386727 D. J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. , 43:525–546, 2001. doi:10.1137/S0036144500378302 C. W. Gardiner. Handbook of Stochastic Methods , 2nd ed. Springer, 1985. http://trove.nla.gov.au/version/46588286

Indirect evolutionary rescue: prey adapts, predator avoids extinction.

Recent studies have increasingly recognized evolutionary rescue (adaptive evolution that prevents extinction following environmental change) as an important process in evolutionary biology and conservation science. Researchers have concentrated on single species living in isolation, but populations in nature exist within communities of interacting species, so evolutionary rescue should also be investigated in a multispecies context. We argue that the persistence or extinction of a focal species can be determined solely by evolutionary change in an interacting species. We demonstrate that prey adaptive evolution can prevent predator extinction in two-species predator–prey models, and we derive the conditions under which this indirect evolutionary interaction is essential to prevent extinction following environmental change. A nonevolving predator can be rescued from extinction by adaptive evolution of its prey due to a trade-off for the prey between defense against predation and population growth rate. As prey typically have larger populations and shorter generations than their predators, prey evolution can be rapid and have profound effects on predator population dynamics. We suggest that this process, which we term ‘indirect evolutionary rescue’, has the potential to be critically important to the ecological and evolutionary responses of populations and communities to dramatic environmental change.

A two-species predator-prey model in an environment enriched by a biotic resource

Classical population growth models assume that the environmental carrying capacity is a fixed parameter, which is not often realistic. We propose a modified predator-prey model where the carrying capacity of the environment is dependent on the availability of a biotic resource. In this model both populations are able to consume the resource, thus altering the environment. Stability, bifurcation and numerical analyses are presented to illustrate the system's dynamical behaviour. Bistability occurs in certain parameter regions. This could describe the transition from a beneficial environment to a detrimental one. We examine special cases of the system and show that both permanence and extinction are possible. References J. Vandermeer. Seasonal isochronic forcing of Lotka Volterra equations. Prog. Theor. Phys. , 96:13–28, 1996. doi:10.1143/PTP.96.13 S. Ikeda and T. Yokoi. Fish population dynamics under nutrient enrichment–-A case of the East Seto Inland Sea. Ecol. Model. , 10:141–165, 1980. doi:10.1016/0304-3800(80)90057-5 S. P. Rogovchenko and Y. V. Rogovchenko. Effect of periodic environmental fluctuations on the Pearl–Verhulst model. Chaos, Solitons, Fractals , 39:1169–1181, 2009. doi:10.1016/j.chaos.2007.11.002 H. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu. A simple model for the total microbial biomass under occlusion of healthy human skin. In Chan, F., Marinova, D. and Anderssen, R.S. (eds) MODSIM2011, 19th International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand., 733–739, 2011. http://www.mssanz.org.au/modsim2011/AA/safuan.pdf P. Meyer and J. H. Ausubel. Carrying capacity: A model with logistically varying limits. Technol. Forecast. Soc. , 61:209–214, 1999. doi:10.1016/S0040-1625(99)00022-0 R. Huzimura and T. Matsuyama. A mathematical model with a modified logistic approach for singly peaked population processes. Theor. Popul. Biol. , 56:301–306, 1999. doi:10.1006/tpbi.1999.1426 J. H. M. Thornley and J. France. An open-ended logistic-based growth function. Ecol. Model. , 184:257–261, 2005. doi:10.1016/j.ecolmodel.2004.10.007 J. H. M. Thornley, J. J. Shepherd and J. France. An open-ended logistic-based growth function: Analytical solutions and the power-law logistic model. Ecol. Model. , 204:531–534, 2007. doi:10.1016/j.ecolmodel.2006.12.026 H. M. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu. Coupled logistic carrying capacity model. ANZIAM J , 53:C172–C184, 2012. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/4972 P. H. Leslie and J. C. Gower. The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika , 47:219–234, 1960. doi:10.1093/biomet/47.3-4.219 B. Basener and D. S. Ross. Booming and crashing populations and Easter Island. SIAM J. Appl. Math. , 65:684–701, 2005. doi:10.1137/S0036139903426952 D. Lacitignola and C. Tebaldi. Symmetry breaking effects on equilibria and time dependent regimes in adaptive Lotka–Volterra systems. Int. J. Bifurcat. Chaos , 13:375–392, 2003. doi:10.1142/S0218127403006595 F. Wang and G. Pang. Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain. Chaos, Solitons, Fractals , 36:1366–1376, 2008. doi:10.1016/j.chaos.2006.09.005 R. Ball. Understanding critical behaviour through visualization: A walk around the pitchfork. Comput. Phys. Commun. , 142:71–75, 2001. doi:10.1016/S0010-4655(01)00322-8 B. Ermentrout. XPP-Aut v. 6.00, 2011. http://www.math.pitt.edu/ bard/xpp/xpp.html R. Malka, E. Shochat and V. R. Kedar. Bistability and bacterial infections. PLOS ONE , 5:1–10, 2010. doi:10.1371/journal.pone.0010010 J. Elf, K. Nilsson, T. Tenson and M. Ehrenberg. Bistable bacterial growth rate in response to antibiotics with low membrane permeability. Phys. Rev. Lett. , 97:258104, 2006. doi:10.1103/PhysRevLett.97.258104 D. Dubnau and R. Losick. Bistability in bacteria. Mol. Microbiol. , 61:564–572, 2006. doi:10.1111/j.1365-2958.2006.05249.x M. Santillan. Bistable behavior in a model of the lac Operon in Escherichia coli with variable growth rate. Biophys. J. , 94:2065–2081, 2008. doi:10.1529/biophysj.107.118026 M. Rosenzweig. Paradox of enrichment: destabilization of exploitation ecosystem in ecological time. Science , 171:385–387, 1971. doi:10.1126/science.171.3969.385

Environmental versus demographic variability in two-species predator-prey models.

We investigate the competing effects and relative importance of intrinsic demographic and environmental variability on the evolutionary dynamics of a stochastic two-species Lotka-Volterra model by means of Monte Carlo simulations on a two-dimensional lattice. Individuals are assigned inheritable predation efficiencies; quenched randomness in the spatially varying reaction rates serves as environmental noise. We find that environmental variability enhances the population densities of both predators and prey while demographic variability leads to essentially neutral optimization.

Analytical Results of a Two-Species Predator-Prey Model

We propose a sequential monomer reaction model for a two-species predator-prey system, in which the aggregates of either species can spontaneously produce or lose one monomer and meanwhile, a type-B aggregate can prey upon one monomer of a type-A aggregate when they meet. Using the mean-field rate equation approach, we analytically investigate the kinetic behavior of the system. The results show that the evolution of the system depends crucially on the details of the rate kernels. The aggregate size distribution of either species approaches the conventional or modified scaling form in most cases. Moreover, the total size of either species grows exponentially with time in some cases and asymptotically retains a constant quantity in other cases, while it decays with time and vanishes finally in the rest cases.

Permanence in Nonautonomous Predator-prey Lotka-Voterra Systems

In this paper some easily verifiable sufficient conditions on the permanence of solutions for general nonautonomous two-species predator-prey model are established. These new criteria improve and extend the results given by Ma, Wang[3], Teng[4] and Ten, Yu[6].

The Influence of Naticid Predation on Evolutionary Strategies of Bivalve Prey: Conclusions from a Model

An age-structured two-species predator-prey model was developed to explore the possible role of selective predation by naticid gastropods in the evolution of the energy allocation of their bivalve prey. The model describes prey fitness in terms of the amount of energy available for reproduction over a lifetime and describes predator fitness in terms of the net energy intake rate of the adult predator. Both species are considered to be age- and size-structured. Predation is size-dependent, and the predator is assumed to select prey in a manner that maximizes its net energy intake. Two evolutionary traits of the prey are considered: the age-dependent allocation of energy between reproduction and growth (X1) and the allocation of energy between overall growth and extra growth in shell thickness (X2). The results of the model are described in terms of prey phenotypic fitness landscapes as functions of predation intensity. These landscapes consist of prey fitness plotted as a function of the parameters, X1 and...