In theoretical ecology, predator–prey interaction is a natural phenomenon that significantly contributes for shaping the community structure and maintaining the ecological diversity. In almost every ...ecological model, the prey species is curtailed by the direct attacking of predator species. However, from different experimental shreds of evidence, it has been observed that fear (felt by prey) for predators can change the physiological behaviour of prey individuals and greatly reduces their reproduction rate as well as enhances their mortality rate. In this current work, we develop and explore a predator–prey model incorporating the cost of perceived fear into the birth and death rates of prey species with Holling type-II functional response. In addition, the intraspecific competition within predator species and a gestation delay are introduced in the model to obtain more realistic and natural dynamics. Feasibility of all the steady states and their stability conditions are analysed in terms of the model parameters. We show that only existence of an interior equilibrium point is sufficient to prevent the extinction of predator species. In this case, either both species can exist together or oscillate around that interior equilibrium point. We can also recognize the parametric region where the system produces multiple coexistence equilibria in which different initial biomass of populations may produce different long-term outcomes. The basic bifurcation analyses of the system exhibit that a higher level of fear or higher intraspecific competition rate helps the population to survive in a coexistence state. For a suitable choice of parametric values, the proposed model may produce the bi-stable phenomenon between two coexistence steady states. We obtain a parametric condition for which the model system experiences a Hopf bifurcation if the delay parameter exceeds some threshold value. All of these theoretical findings are verified by various numerical examples.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPUK, ZAGLJ, ZRSKP
Two types of boundedly rational monopolists are studied when the marginal revenue is not necessarily negative sloping. A limited monopolist (ℓ-monopolist) knows only the price and output values in ...two previous periods. A knowledgeable monopolist (k-monopolist) knows the analytic expression of the marginal profit function and forms sales expectations based on past market information. The k-monopolist adjusts its output levels according to the usual gradient process, while the ℓ-monopolist approximates the marginal profit with a two-point finite difference formula. Stability conditions and complex dynamics are studied in discrete and continuous time scales. In the discrete model, the two monopolists exhibit different dynamics after the stability loss. The ℓ-monopolist has a cascade of a Neimark-Sacker bifurcation. In contrast, the k-monopolist has two different routes, a period-doubling and a Neimark-Sacker bifurcation, depending on how to form sales expectations. In the continuous case, the discrete models are transformed into continuous models via Euler transformation. The stability switching curves are analytically constructed and the directions of stability switching are characterized by computing the stability index for each point of the curves. It is numerically demonstrated that two monopolists exhibit similar dynamics and the k-monopolist approximates the ℓ-monopolist under a special circumstance.
•The discrete-time monopoly model of Tonu Puu 3 is converted to a two-delay differential model via Euler transformation.•The stability switching and its directions are analytically obtained and then numerically confirmed.•A slow speed of adjustment leads the monopolists to a stable state and a rapid speed could be a source of complex oscillations.•The ℓ-monopolist's dynamic behavior can be approximated by the k-monopolist's, although the two monopolists seem different.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPUK, ZAGLJ, ZRSKP
Present article has proposed a general eco-epidemic model with disease in predator population subject to Beddington–DeAngelis type incidence rate with the significance of fear in prey individuals and ...also analyzed the impact of external food supply to the sound predator. As an additional factor in representing the interplay amongst the predator and prey species, the Holling type-II functional response in the context of intra-specific competition within the prey species is taken into consideration. The model dynamics is studied by employing discrete time-delay in prey and gestation delay in predator in order to generate more authentic and natural dynamics. Positivity, uniform boundedness, and uniform persistence of the solutions for the model system have been explained analytically. Furthermore, the extinction criteria of the predator population have been explored and also illustrated by numerical simulations. For the non-delayed model, the existence and stability conditions of all conceivable critical points are investigated associated with the model parameters. The basic fundamental bifurcation assessments of the model reveal the formation of local bifurcations (Hopf-bifurcation and transcritical bifurcation) and provide the parametric region for occurrence of Bautin bifurcation and Gavrilov–Guckenheimer bifurcation. It is also reported that the provision of supplementary food to the sound predator may enable to remove the infection more promptly. Afterward, the stability dynamics of the coexistence state for different configurations of the delay factors have been scrutinized, as well as the delay parameters may produce oscillations via Hopf-bifurcation if they exceed some threshold value. But, when the gestation delay has been incorporated only in infected predators, no critical point can be obtained at which the Hopf-bifurcation takes place for the considered parameter values. Most of the theoretical discoveries have been certified through several numerical experiments constructed with the application of MATLAB and MATCONT.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPUK, ZAGLJ, ZRSKP
A modified Holling type II (variable predator search rate rather than constant) predator–prey interaction has been proposed where apart from direct consumption the prey biomass is affected by the ...fear of predator. The goal of this work is to analyze the dynamic nature of the system when the constant search rate in the Holling type II response is replaced by a prey-dependent search rate in the presence of fear effect which raises the anti-predator behaviour of prey. Some basic properties such as positivity, uniform boundedness, and persistence have been provided. Next, we have studied the existing conditions and stability criteria of all equilibria. Then, the conditions for the occurrence of local bifurcations of co-dimension 1 are given. Furthermore, we have investigated the effects of all the combinations of delay factors (breeding and gestation delay) for which the proposed system switches its stability through supercritical Hopf bifurcation. Numerical simulations are conducted to validate analytical results. It is observed that level of fear can not only reduce predator’s growth at the interior equilibrium but also stabilize the system by excluding the existence of a limit cycle. We also discuss the similarities and differences between our proposed system and the usual Holling type II model. Lastly, the deterministic system has been extended to a stochastic model using nonlinear perturbation.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
There have been some recent attempts to combine Cournot and Bertrand duopolies in one single model. Unfortunately, these attempts do not work. A commodity cannot be homogenous and non-homogenous at ...the same time. It is always the consumers, who decide whether they perceive competing products as identical or as different brands for which they are willing to pay different prices. There is, of course, nothing that forbids the coexistence of both such consumer groups. Neither is there any obstacle for the competing sellers to sell to both markets. Then we only need an old idea from economic theory, i.e., price discrimination, to rectify the logic. By this the challenging combination idea comes on a stable footing. The model also results in some interesting mathematical facts, such as mulistability and coexistence of attractors.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPUK, ZAGLJ, ZRSKP
In this work, we have investigated a phytoplankton–zooplankton model system such that the toxic liberation by toxic producing phytoplankton obeys a discrete time lag. Here, we have taken modified ...Holling type II functional response, based on the fact that search rate of zooplankton depends on the biomass of phytoplankton, rather than constant. The similarities but also important differences between our studied model and Holling type II system are discussed. The basic dynamical properties of the proposed model have been studied briefly in absence of delay factor. Next, we have analyzed the dynamic nature of the delayed system and also found the existence of stability switching phenomena. Numerical simulations are conducted to validate our analytical findings using MATLAB. Numerically, the phenomena of bistability has been recognized. Lastly, our proposed deterministic system has been compared with a stochastic model using Gaussian white noise terms due to the effect of environmental fluctuations.
•Formulates a toxic producing phytoplankton–zooplankton system in which toxic liberation requires a discrete time lag.•It assumes that zooplankton’s search rate depends on the biomass of phytoplankton species.•Similarities and important differences between the proposed system and Holling type II model are discussed.•Pictorial scenarios have been illustrated using MATLAB to verify the analytical findings.•The effect of environmental fluctuations on the planktonic model has been discussed numerically.
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GEOZS, IJS, IMTLJ, KILJ, KISLJ, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, UILJ, UL, UM, UPUK, ZAGLJ, ZRSKP
A field experiment on terrestrial vertebrates observes that direct predation on predator–prey interaction can not only affect the population dynamics but the indirect effect of predator’s fear (felt ...by prey) through chemical and/or vocal cues may also reduce the reproduction of prey and change their life history. In this work, we have described a predator–prey model with Holling type II functional response incorporating prey refuge. Irrespective of being considering either a constant number of prey being refuged or a proportion of the prey population being refuged, a different growth rate and different carrying capacity for the prey population in the refuge area are considered. The total prey population is divided into two subclasses: (i) prey
in the refuge area and (ii) prey
in the predatory area. We have taken the migration of the prey population from refuge area to predatory area. Also, we have considered a benefit from the antipredation response of the prey population
in presence of cost of fear. Feasible equilibrium points of the proposed system are derived, and the dynamical behavior of the system around equilibria is investigated. Birth rate of prey in predatory region has been regarded as bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighborhood of the interior equilibrium point. Moreover, the conditions for occurrence of transcritical bifurcations have been determined. Further, we have incorporated discrete-type gestational delay on the system to make it more realistic. The dynamical behavior of the delayed system is analyzed. Finally, some numerical simulations are given to verify the analytical results.
In this paper we analyze a duopolistic market with heterogeneous firms when the demand function is isoelastic (Puu, T., 1991. Chaos in duopoly pricing. Chaos, Solitons and Fractals 1, 573–581.). We ...consider the same heterogeneous firms of Zhang et al. (Zhang, J., Da, Q., Wang, Y., 2007. Analysis of nonlinear duopoly game with heterogeneous players. Economic Modelling 24, 138–148.) introducing a nonlinearity in the demand function instead of the cost function. Stability conditions of the Nash equilibrium and complex dynamics are studied. In particular we show two different routes to complicated dynamics: a cascade of flip bifurcations leading to periodic cycles (and chaos) and the Neimark-Sacker bifurcation which originates an attractive invariant closed curve. Comparisons with respect to the Puu model and the model of Zhang et al. are performed.
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Interaction between prey and predator is a natural phenomenon in ecology that significantly contributes to the structure of ecological variety. Recent studies indicate that the presence of predator ...can influence the physiological behaviour of prey species to such an extent that it can be more efficient than direct predation in decreasing the prey biomass. Moreover, such non-lethal effects can be carried over through seasons or generations. In this present article, we analyze the impact of predator-induced fear and its carry-over effect in a predator-prey model in which the predator species can access some alternative or additional food sources. Well-posedness of the system and some basic dynamical properties such as extinction criteria, stability analysis with global stability, uniform persistence etc. are discussed thoroughly. From the bifurcation analyzes, we can observe that fear and its carry-over effect can not switch the stability from one equilibrium state to other equilibrium state. However once the coexistence equilibrium state occurs in the system, a higher level of fear can stabilize it. On the other hand, higher level of carry-over effect promote the oscillatory dynamics around the coexistence state. Therefore, fear and its carry-over effects have two opposite roles in the stability of the coexistence equilibrium. Predator species may go extinct if the effective quantity of additional food is sufficiently low. Next we study the model system in presence of gestation delay and observe some interesting dynamics by taking the delay parameter as a bifurcation parameter. Our study demonstrates how non-lethal effects alter the dynamics of a prey-predator model and provides valuable biological insights, particularly into the dynamics of small food web.
In this paper, taking the factor of product service provided by the manufacturers into consideration, a static Bertrand duopoly game with service factor is studied first, in which these two oligarchs ...produce differentiated products. A dynamic Bertrand duopoly game with bounded rationality is established by using the gradient mechanism. Keeping the adjustment speeds in a relatively small range may help the long-term stable operation of the market. It is found that there is another 2-cycle, different from the one flip bifurcated from the fixed point, which may appear through a saddle-node bifurcation. The unstable set of the saddle cycle, connecting the saddle to the node, gives a closed invariant curve. In addition, the emergence of intermittent chaos implies that the established economic system has the capability of self-regulating, where PM-I intermittency and crisis-induced intermittency have been studied. With the help of the critical curves, the qualitative changes on the basin of attraction are investigated.
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EMUNI, FIS, FZAB, GEOZS, GIS, IJS, IMTLJ, KILJ, KISLJ, MFDPS, NLZOH, NUK, OILJ, PNG, SAZU, SBCE, SBJE, SBMB, SBNM, UKNU, UL, UM, UPUK, VKSCE, ZAGLJ
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