Constant Rate Harvesting Research Articles

Dynamics of a Leslie-Gower predator-prey system with hunting cooperation and prey harvesting

<p style='text-indent:20px;'>In this paper, we study the dynamics of a Leslie-Gower predator-prey system with hunting cooperation among predator population and constant-rate harvesting for prey population. It is shown that there are a weak focus of multiplicity up to three and a cusp of codimension at most two for various parameter values, and the system exhibits two saddle-node bifurcations, a Bogdanov-Takens bifurcation of codimension two and a Hopf bifurcation as the bifurcation parameters vary. The results developed in this article reveal far more complex dynamics compared to the Leslie-Gower system and show how the prey harvesting and the hunting cooperation affect the dynamics of the system. In particular, there exist some critical values of prey harvesting and hunting cooperation such that the predator and prey populations are at risk of extinction if the intensities of harvesting and hunting cooperation are greater than these critical values. Moreover, numerical simulations are presented to illustrate our theoretical results.</p>

Dynamical Behaviors of a Fractional-Order Three Dimensional Prey-Predator Model

In this paper, the dynamical behavior of a three-dimensional fractional-order prey-predator model is investigated with Holling type III functional response and constant rate harvesting. It is assumed that the middle predator species consumes only the prey species, and the top predator species consumes only the middle predator species. We also prove the boundedness, the non-negativity, the uniqueness, and the existence of the solutions of the proposed model. Then, all possible equilibria are determined, and the dynamical behaviors of the proposed model around the equilibrium points are investigated. Finally, numerical simulations results are presented to confirm the theoretical results and to give a better understanding of the dynamics of our proposed model.

Fish harvesting strategies with allee effect

Fish harvesting is important for nutritious food availability in human diet. In this manuscript, a simple fish harvesting model is developed using logistic growth along with allee effect. We study the basic mathematical properties of the model as equilibrium analysis and stability. For better understanding of the model performance, we perform the simulation of the system with different constant harvesting rate and periodic harvesting rate. The results shows that periodic harvesting is economically beneficial than the constant harvesting in fish stock.

Bifurcations of a Leslie‐Gower prey‐predator system with ratio‐dependent Holling IV functional response and prey harvesting

The dynamics of a Leslie‐Gower prey‐predator system with ratio‐dependent Holling IV functional response and constant harvesting rate of prey are taken into account. The results developed in this article reveal far richer dynamics compared with the system without harvesting. We first make qualitative and bifurcation analysis of the system without harvesting and show that the system has a weak focus of multiplicity at most 2, at which a Hopf bifurcation occurs. However, the system with harvesting has four nonhyperbolic equilibria for some parameter values, such as two saddle‐node, a cusp, and a weak focus of multiplicity at most 4, and exhibits two saddle‐node bifurcations, a Bogdanov‐Takens bifurcation of codimension 2, and a Hopf bifurcation. It reveals that there exist some critical harvesting values such that the species are in danger of extinction when the harvesting rate is greater than the critical values, which indicates that the dynamics of the system are sensitive to the constant prey harvesting. Moreover, numerical simulations are presented to illustrate our theoretical results.

Stability and Bifurcation for a Single‐Species Model with Delay Weak Kernel and Constant Rate Harvesting

In this paper, we consider the effect of constant rate harvesting on the dynamics of a single‐species model with a delay weak kernel. By a simple transformation, the single‐species model is transformed into a two‐dimensional system. The existence and the stability of possible equilibria under different conditions are carried out by analysing the two‐dimensional system. We show that there exists a critical harvesting value such that the population goes extinct in finite time if the constant rate harvesting u is greater than the critical value, and there exists a degenerate critical point or a saddle‐node bifurcation when the constant rate harvesting u equals the critical value. When the constant rate harvesting u is less than the critical value, sufficient conditions about the existence of the Hopf bifurcation are derived by topological normal form for the Hopf bifurcation and calculating the first Lyapunov coefficient. The key results obtained in the present paper are illustrated using numerical simulations. These results indicate that it is important to select the appropriate constant rate harvesting u.

Local stability analysis on Lotka‐Volterra predator‐prey models with prey refuge and harvesting

We propose a predator‐prey model by incorporating a constant harvesting rate into a Lotka‐Volterra predator‐prey model with prey refuge. All the positive equilibria and the local stability of the proposed model are studied and analyzed by sorting out the intervals of the parameters involved in the model. These intervals of the parameters exhibit the effects on the dynamical behaviors of prey and predators. The emphasis is put on the ranges of the prey refuge constant and harvesting rate. We show that the model has three‐type positive boundary equilibria and one positive interior equilibrium. By using the qualitative theory for planar systems, we show that the three‐type boundary positive equilibria can be saddles, saddle nodes, topological saddles, or stable or unstable nodes, and the interior positive equilibrium is locally asymptotically stable. Under suitable restrictions on the parameters, we prove that the positive interior equilibrium is a stable node. It remains open that under what conditions on the parameters is the positive interior equilibrium a focus.

Periodic solution and control optimization of a prey-predator model with two types of harvesting

In this work, a prey-predator model with both state-dependent impulsive harvesting and constant rate harvesting is investigated, where the replenishment rate of prey and the harvesting rate are linearly related with the selected threshold. By first using the successor function method and differential equation geometry theory, the existence, uniqueness and asymptotic stability of the order-1 periodic solution are discussed. And then numerical simulations with an example are given to illustrate the feasibility of the theorem-related results. Moreover, in order to increase the total profit, the optimization strategy is presented and the optimal threshold is obtained.

The Dynamics of Discrete System with Constant Rate Harvesting

The Dynamics of Discrete System with Constant Rate Harvesting

Dynamics of a Beddington-DeAngelis type predator-prey system with constant rate harvesting

Dynamics of a Beddington-DeAngelis type predator-prey system with constant rate harvesting

Analysis of a Holling II system with frequency-dependent fitness and constant harvesting rate of prey

Analysis of a Holling II system with frequency-dependent fitness and constant harvesting rate of prey

Optimal harvesting of a spatial renewable resource

In this paper we investigate the optimal harvesting of a renewable natural resource. While in most standard approaches the resource is located at a single point, we allow the resource to be distributed spatially. Consequently, an agent who exploits the resource has to travel from one location to another. For a fixed planning horizon, we investigate the speed and the path of harvesting chosen by the agent. We show that the agent adjusts this speed so as to visit each location only once, even in the absence of travelling cost. Since the agent does not return to any location for a second harvest, it is optimal to fully deplete the resource upon arrival. A similar type of bang–bang solution results when we drop the assumption of a constant harvesting rate: allowing for a variable harvesting rate, the agent chooses to fully exploit the resource either in the last or in the first travelling period. A society interested in conserving some of the resource thus has to take measures to limit the exploitative behaviour of the agent.

Bifurcation analysis of a predator-prey system with self- and cross-diffusion and constant harvesting rate

In this paper, we focus on a ratio dependent predator-prey system with self- and cross-diffusion and constant harvesting rate. By making use of a brief stability and bifurcation analysis, we derive the symbolic conditions for Hopf, Turing and wave bi- furcations of the system in a spatial domain. Additionally, we illustrate spatial pattern formations caused by these bifurcations via numerical examples. A series of numerical examples reveal that one can observe several typical spatiotemporal patterns such as spotted, spot-stripelike mixtures due to Turing bifurcation and an oscillatory wave pat- tern due to the wave bifurcation. Thus the obtained results disclose that the spatially extended system with self-and cross-diffusion and constant harvesting rate plays an important role in the spatiotemporal pattern formations in the two dimensional space.

Defoliation and bark harvesting affect life‐history traits of a tropical tree

Summary Selectively harvesting whole individuals in managed populations (e.g. fisheries, hunting) has substantial effects on life expectancy and age at maturity. Although demographic rates of trees are impacted by recurrent harvest of plant organs (e.g. fruit, leaf, bark) known as non‐timber forest products, the effect of such harvesting on life‐history traits is less explored. Here, we investigate how different strategies of foliage and bark harvest by local people affect life expectancy and age at maturity of Khaya senegalensis across two climatic regions in West Africa. We compare elasticities of life expectancy to perturbation of vital rates to the elasticities of population growth rate, emphasizing how the two kinds of elasticity address distinct biological issues and management goals. Life expectancy was shorter and reproduction delayed in the dry than in the moist region, indicating a cost of drought to life‐history traits. Harvesting at constant rates only affects (increased) life expectancy in the moist region and (reduced) age at first reproduction in the dry region. Models in which harvest intensity varies stochastically over time show results similar to those with constant harvesting rate. However, there is an interactive effect of harvest on life‐history traits between regions. Increasing harvesting pressure accelerates maturity and reduces life expectancy in the dry region but delays reproduction and increases life expectancy in the moist region. This indicates that stochasticity magnifies the effect of harvesting on life‐history traits. With the notable exception of heavily harvested populations in the moist region, life expectancy is more elastic to survival of early rather than late life stages. This pattern contrasts with the elasticity of population growth rate; the latter is more elastic to survival of late rather than early life stages. Synthesis. Our results suggest that plant harvesting can affect life‐history traits but in different ways across an environmental gradient. Failure to account for stochasticity in harvesting rate can mask some of these effects. Our results also indicate that processes driving plant life expectancy, at least for long‐lived species, may differ from those driving population dynamics.

Periodic solutions and homoclinic bifurcation of a predator–prey system with two types of harvesting

In this paper, a predator–prey model with both constant rate harvesting and state dependent impulsive harvesting is analyzed. By using differential equation geometry theory and the method of successor functions, the existence, uniqueness and stability of the order one periodic solution have been studied. Sufficient conditions which guarantee the nonexistence of order k (k≥2) periodic solution are given. We also present that the system exhibits the phenomenon of homoclinic bifurcation under some parametric conditions. Finally, some numerical simulations and biological explanations are given.

Bogdanov-Takens Bifurcation of a Delayed Ratio-Dependent Holling-Tanner Predator Prey System

A delayed predator prey system with refuge and constant rate harvesting is discussed by applying the normal form theory of retarded functional differential equations introduced by Faria and Magalhães. The analysis results show that under some conditions the system has a Bogdanov-Takens singularity. A versal unfolding of the system at this singularity is obtained. Our main results illustrate that the delay has an important effect on the dynamical behaviors of the system.

The saddle-node-transcritical bifurcation in a population model with constant rate harvesting

We study the interaction of saddle-node and transcritical bifurcations in a Lotka-Volterra model with a constant term representing harvesting or migration. Because some of the equilibria of the model lie on an invariant coordinate axis, both the saddle-node and the transcritical bifurcations are of codimension one. Their interaction can be associated with either a single or a double zero eigenvalue. We show that in the former case, the local bifurcation diagram is given by a nonversal unfolding of the cusp bifurcation whereas in the latter case it is a nonversal unfolding of a degenerate Bogdanov-Takens bifurcation. We present a simple model for each of the two cases to illustrate the possible unfoldings. We analyse the consequences of the generic phase portraits for the Lotka-Volterra system.

BIFURCATIONS OF A HOLLING-TYPE II PREDATOR–PREY SYSTEM WITH CONSTANT RATE HARVESTING

The objective of this paper is to study the dynamical properties of a Holling-type II predator–prey system with constant rate harvesting. It is shown that the model has at most three equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the saddle-node bifurcation, the degenerate Bogdanov–Takens bifurcation of codimension 3, the supercritical and subcritical Hopf bifurcation, the generalized Hopf bifurcation. These results reveal far richer dynamics than that of the model with no harvesting.

Hopf bifurcation and non-hyperbolic equilibrium in a ratio-dependent predator–prey model with linear harvesting rate: Analysis and computation

In this paper, we propose a ratio-dependent predator–prey model with linear harvesting rate, which can describe the real predator–prey system better than the previously published model with constant harvesting in [D. Xiao, L.S. Jennings, Bifurcation of a ratio-dependent predator–prey system with constant rate harvesting, SIAM J. Appl. Math. 65 (3) (2005) 737–753]. We show the different types of system behaviors realized for various parameter values. In particular, there exist areas of coexistence (which may be steady or oscillating), areas in which both populations become extinct, and areas of “conditional coexistence” depending on the initial values. By choosing the modulus of linear harvesting rate as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. An explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory. One of the main mathematical features of ratio-dependent models, distinguishing this class from other predator–prey models, is that the origin is a non-hyperbolic equilibrium, whose characteristics crucially determine the main properties of the model. The complete analysis of possible topological structures in a neighborhood of the origin, as well as asymptotics to orbits tending to this point, is given. Experimental motivation is made to summarize the effect of harvesting on coexistence and extinction.

Dynamics of a Ivlev-type predator–prey system with constant rate harvesting

In this paper, by using the analysis of qualitative method and bifurcation theory, we investigate the dynamical properties of the Ivlev-type predator–prey model with nonzero constant prey harvesting and with or without time delay, respectively. It is shown that the system we considered can exhibit the subcritical and supercritical Hopf bifurcation. We also study the effect of the time delay on the dynamics of the system. By choosing the delay τ as a bifurcation parameter, we show that Hopf bifurcation can occur as the delay τ crosses some critical values. The direction and stability of the Hopf bifurcation are investigated by following the procedure of deriving normal form given by Faria and Magalhães. Finally, numerical simulations are performed to illustrate the obtained results.

Numerical bifurcation of predator-prey fractional differential equations with a constant rate harvesting

In this article saddle and Hopf bifurcation points of predator-prey fractional differential equations system with a constant rate harvesting are investigated. The numerical results based on Grunwald-Letnikov discretization for fractional differential equations together with the Mickens' non-standard discretization method agree with those found by the corresponding ordinary differential equation system.