Beverton-Holt Equation Research Articles

THE BEHAVIOR OF A CLASS HOST-PARASITOID MODELS WITH HOST REFUGE AND STRONG ALLEE EFFECT UPON THE HOST POPULATION

This paper studies the dynamics of a class of host-parasitoid models with host refuge and the strong Allee effect upon the host population. Without the parasitoid population, the Beverton–Holt equation governs the host population. The general probability function describes the portion of the hosts that are safe from parasitism. The existence and local behavior of solutions around the equilibrium points are discussed. We conclude that the extinction equilibrium will always have its basin of attraction which implies that the addition of the host refuge will not save populations from extinction. By taking the host intrinsic growth rate as the bifurcation parameter, the existence of the Neimark–Sacker bifurcation can be shown. Finally, we present numerical simulations to support our theoretical findings.

On an Extended Time-Varying Beverton–Holt Equation Subject to Harvesting Monitoring and Population Excess Penalty

This paper considers a more general eventually time-varying Beverton–Holt equation for species evolution which can include a harvesting action and a penalty for overpopulation numbers. The harvesting action may be positive (typically consisting of hunting or fishing) or negative which refers to repopulation within the environment. One considers also a penalty of quadratic type on the overpopulation and the introduction of a term related to Allee effect to take account of small levels of population. The intrinsic growth rate is assumed either to exceed unity or to be under unity. In the second case, the extinction point is a locally stable attractor while the other positive equilibrium point is unstable contrarily to the commonly studied case of intrinsic growth rate exceeding unity where the above roles are inverted. This consequence implies that the extinction point is also globally asymptotically stable for any given finite initial condition. In the case when the eventual overpopulation is penalized with a sufficiently large coefficient which exceeds a prescribed threshold, to quantify such an excess, only a globally asymptotically stable extinction attractor is present and no other positive equilibrium points exist. In the case of a positive moderate quadratic evaluation term for such an overpopulation, one or two positive equilibrium points coexist with the extinction one. The smaller one is unstable contrarily to the extinction equilibrium which is locally asymptotically stable. If it exists a second largest positive equilibrium point, being distinct to the above-given one, then it can be unstable or locally stable depending on the parameterization. Also, some methods of monitoring the population evolution through control laws on the harvesting action are discussed.

The Beverton-Hold model on isolated time scales.

In this work, we formulate the Beverton-Holt model on isolated time scales and extend existing results known in the discrete and quantum calculus cases. Applying a recently introduced definition of periodicity for arbitrary isolated time scales, we discuss the effects of periodicity onto a population modeled by a dynamic version of the Beverton-Holt equation. The first main theorem provides conditions for the existence of a unique ω -periodic solution that is globally asymptotically stable, which addresses the first Cushing-Henson conjecture on isolated time scales. The second main theorem concerns the generalization of the second Cushing-Henson conjecture. It investigates the effects of periodicity by deriving an upper bound for the average of the unique periodic solution. The obtained upper bound reveals a dependence on the underlying time structure, not apparent in the classical case. This work also extends existing results for the Beverton-Holt model in the discrete and quantum cases, and it complements existing conclusions on periodic time scales. This work can furthermore guide other applications of the recently introduced periodicity on isolated time scales.

Global Dynamics of Delayed Sigmoid Beverton–Holt Equation

In this paper, certain dynamic scenarios for general competitive maps in the plane are presented and applied to some cases of second-order difference equation xn+1=fxn,xn−1, n=0,1,…, where f is decreasing in the variable xn and increasing in the variable xn−1. As a case study, we use the difference equation xn+1=xn−12/cxn−12+dxn+f, n=0,1,…, where the initial conditions x−1,x0≥0 and the parameters satisfy c,d,f>0. In this special case, we characterize completely the global dynamics of this equation by finding the basins of attraction of its equilibria and periodic solutions. We describe the global dynamics as a sequence of global transcritical or period-doubling bifurcations.

Risk sensitivity in Beverton–Holt fishery with multiplicative harvest

AbstractWe present a steady‐state threshold risk analysis framework for exploited populations following the Beverton–Holt recurrence. The Beverton–Holt model is widely applied in the assessment of species biomass and fitted to experimental data to obtain a suitable range of parameter values. To account for the uncertainty in these parameter values, such as the growth rate, we analyze the probability of the steady‐state harvested population falling below a critical threshold. More precisely, the Beverton–Holt equation with constant multiplicative survival, constant carrying capacity, and constant growth rate is considered. Under the assumption of a stochastic distributed proliferation rate, we analyze the risk of the steady‐state population falling below a specified threshold, under constant harvest. We demonstrate a characteristic sensitivity property of that risk, that we refer to as “instability wedge,” in the parameter space. We continue the study by assuming a 2‐periodic carrying capacity, representing seasonal changes in the population's environment. Again, we demonstrate the risk sensitivity phenomenon. Although the growth rate was chosen to have a uniform distribution in presented examples, the framework extends to other probability distributions.

Almost periodic stochastic Beverton-Holt difference equation with higher delays and with competition between overlapping generations

Abstract The paper studies the existence of an almost periodic solution of some system of stochastic Beverton-Holt equation with higher delays and with competition between overlapping generations under some reasonable assumptions.

Neimark–Sacker Bifurcation and Stability of a Certain Class of Host-Parasitoid Models with Host Refuge Effect

In this paper, we consider the dynamics of a certain class of host-parasitoid models, where some hosts are completely free from parasitism either with or without a spatial refuge and the host population is governed by the Beverton–Holt equation. We assume that, in each generation, a constant portion of the host population may find a refuge and be safe from the attack by parasitoids. We derive some criteria for the Neimark–Sacker bifurcation. Then, we apply the developed theory to the three well-known cases: [Formula: see text] model, Hassel and Varley model, and parasitoid–parasitoid model. Intensive numerical calculations suggest that the last two models undergo a supercritical Neimark–Sacker bifurcation.

Attenuation in the almost periodic Beverton-Holt equation

ABSTRACTIt is known that the Beverton-Holt equation with periodically varying carrying capacity has a globally attracting solution and the solution exhibits attenuation, i.e. the average of the solution over one period is strictly less than the average of the carrying capacity. Interpreted this means a periodically varying environment has a deleterious effect on the average of the solution. Also known is a randomly varying carrying capacity also yields attenuation. In this work the authors show that an almost periodic carrying capacity also yields attenuation.

Influence of technological progress and renewability on the sustainability of ecosystem engineers populations.

Overpopulation and environmental degradation due to inadequate resource-use are outcomes of human's ecosystem engineering that has profoundly modified the world's landscape. Despite the age-old concern that unchecked population and economic growth may be unsustainable, the prospect of societal collapse remains contentious today. Contrasting with the usual approach to modeling human-nature interactions, which are based on the Lotka-Volterra predator-prey model with humans as the predators and nature as the prey, here we address this issue using a discrete-time population dynamics model of ecosystem engineers. The growth of the population of engineers is modeled by the Beverton-Holt equation with a density-dependent carrying capacity that is proportional to the number of usable habitats. These habitats (e.g., farms) are the products of the work of the individuals on the virgin habitats (e.g., native forests), hence the denomination engineers of ecosystems to those agents. The human-made habitats decay into degraded habitats, which eventually regenerate into virgin habitats. For slow regeneration resources, we find that the dynamics is dominated by rounds of prosperity and collapse, in which the population reaches vanishing small densities. However, increase of the efficiency of the engineers to explore the resources eliminates the dangerous oscillatory patterns of feast and famine and leads to a stable equilibrium that balances population growth and resource availability. This finding supports the viewpoint of growth optimists that technological progress may avoid collapse.

Comparative Performance of Three Length‐Based Mortality Estimators

AbstractLength‐based methods provide alternatives for estimating the instantaneous total mortality rate (Z) in exploited marine populations when data are not available for age‐based methods. We compared the performance of three equilibrium length‐based methods: the length‐converted catch curve (LCCC), the Beverton–Holt equation (BHE), and the length‐based spawning potential ratio (LB‐SPR) method. The LCCC and BHE are two historically common procedures that use length as a proxy for age. From a truncated length‐frequency distribution of fully selected animals, the LCCC estimates Z with a regression of the logarithm of catch at length by the midpoint of the length‐bins, while the BHE estimates Z as a function of the mean length. The LB‐SPR method is a likelihood‐based population dynamics model, which—unlike the LCCC and BHE—does not require data truncation. Using Monte Carlo simulations across a range of scenarios with varying mortality and life history characteristics, our study showed that neither the LCCC nor the BHE was uniformly superior in terms of bias or root mean square error across simulations, but these estimators performed better than LB‐SPR, which had the largest bias in most cases. Generally, if the ratio of natural mortality (M) to the von Bertalanffy growth rate parameter (K) is low, then the BHE is most preferred, although there is likely to be high bias and low precision. If M/K is high, then the LCCC and BHE performed better and similarly to each other. Differences in performance among commonly used truncation methods for the LCCC and BHE were small. The LB‐SPR method did not perform as well as the classical methods but may still be of interest because it provides estimates of a logistic selectivity curve. The M/K ratio provided the most contrast in the performance of the three methods, suggesting that it should be considered for predicting the likely performance of length‐based mortality estimators.

A nonautonomous Beverton–Holt equation of higher order

In this paper, we discuss a certain nonautonomous Beverton–Holt equation of higher order. After a brief introduction to the classical Beverton–Holt equation and recent results, we solve the higher-order Beverton–Holt equation by rewriting the recurrence relation as a difference system of order one. In this process, we examine the existence and uniqueness of a periodic solution and its global attractivity. We continue our analysis by studying the corresponding second Cushing–Henson conjecture, i.e., by relating the average of the unique periodic solution to the average of the carrying capacity.

Optimal harvesting policy for the Beverton-Holt quantum difference model

In this paper, we introduce exploitation to the Beverton-Holt equation in the quantum calculus time setting. We first give a brief introduction to quantum calculus and to the Beverton-Holt q-difference equation before formulating the harvested Beverton-Holt q-difference equation. Under the assumption of a periodic carrying capacity and periodic inherent growth rate, we derive its unique periodic solution, which globally attracts all solutions. We further derive the optimal harvest effort for the Beverton-Holt q-difference equation under the catch-per-effort hypothesis. Examples are provided and discussed in the last section.

Resonance and attenuation in the n-periodic Beverton–Holt equation

An exact expression is derived relating the state average of the periodic solution to the average of the environmental carrying capacities for the periodic Beverton–Holt equation for arbitrary period. By studying numerically period 3 case, we show that the correlation coefficient of the intrinsic growth rates and , is not relevant in determining attenuation or resonance. By studying period 4 case, it is shown that if the intrinsic growth rate jumps upward along with steadily increasing carrying capacities, then resonance prevails. A period 7 example using out-of-step step functions is also seen to produce resonance.

The Beverton–Holt q-difference equation

The Beverton–Holt model is a classical population model which has been considered in the literature for the discrete-time case. Its continuous-time analogue is the well-known logistic model. In this paper, we consider a quantum calculus analogue of the Beverton–Holt equation. We use a recently introduced concept of periodic functions in quantum calculus in order to study the existence of periodic solutions of the Beverton–Holt q-difference equation. Moreover, we present proofs of quantum calculus versions of two so-called Cushing–Henson conjectures.

Difference equations with the Allee effect and the periodic Sigmoid Beverton–Holt equation revisited

In this paper, we investigate the long-term behaviour of solutions of the periodic Sigmoid Beverton–Holt equation where the a n and δ n are p-periodic positive sequences. Under certain conditions, there are shown to exist an asymptotically stable p-periodic state and a p-periodic Allee state with the property that populations smaller than the Allee state are driven to extinction while populations greater than the Allee state approach the stable state, thus accounting for the long-term behaviour of all initial states. This appears to be the first study of the equation with variable δ. The results are discussed with possible interpretations in Population Dynamics with emphasis on fish populations and smooth cordgrass.

ON THE IMPULSIVE BEVERTON–HOLT EQUATION AND EXTINCTION CONDITIONS IN POPULATION DYNAMICS

Abstract This paper is devoted to investigate extinction and nonextinction conditions of the extended Beverton–Holt equation (BHE) for dynamics of populations in Ecology when potential discontinuities at sampling points are included in the model. The proposed model is described by means of four sequences of parameters. Two of them are the intrinsic growth rate and the carrying capacity sequences which are included in the basic BHE model. The other two ones, namely, the harvesting (i.e., the hunting or fishing quota) and the internal consumption (which can include positive and negative migrations in the considered population habitat) sequences are included to parameterize the model discontinuities. Such discontinuities are related to impulses in the corresponding continuous‐time logistic equation. The obtained results establish how the harvesting quota and/or the internal consumption has to be fixed to guarantee the population nonextinction or, eventually, its extinction. Finally, some controllability results related to the search of a carrying capacity sequence such that the solution of the proposed impulsive BHE tracks a reference one are obtained.

Control issues for the Beverton–Holt equation in ecology by locally monitoring the environment carrying capacity: Non-adaptive and adaptive cases

This paper proposes a control algorithm to govern the solution of the Beverton–Holt equation (BHE) under the potentially presence of additive disturbances. The BHE to be controlled is defined by certain intrinsic growth rate and environment carrying capacity sequences, the last one being susceptible of local modifications around nominal values. In fact, the control action provides the carrying capacity which makes that the solution of the current BHE tracks a reference sequence given by another BHE defined by appropriate intrinsic growth rate and environment carrying capacity sequences. In this context, the fact that the inverse of the BHE is a discrete time-varying linear system is taken into account where the inverse of the carrying capacity sequence plays the role of control sequence. The current and the reference BHEs have to be close enough to each other in order that local modifications of the carrying capacity be able to meet the tracking objective. A feedback control law is designed to achieve such an objective with a zero tracking-error in the ideal case of known intrinsic growth rate sequence and no presence of disturbances. An adaptive control law, with the associated parameter estimation algorithm, is considered when the intrinsic growth rate is fully or partially unknown and disturbances are present. Such a control strategy guarantees a bounded tracking-error with the error converging asymptotically to zero in case that additive disturbances also converge to zero. Some results obtained from a simulation example illustrate the effectiveness of this control strategy.

On the stochastic Beverton–Holt equation with survival rates

The paper studies a Beverton–Holt difference equation, in which both the recruitment function and the survival rate vary randomly. It is then shown that there is a unique invariant density, which is asymptotically stable. Moreover, a basic theory for random mean almost periodic sequence on is given. Then, some sufficient conditions for the existence of a mean almost periodic solution to the stochastic Beverton–Holt equation are given.

About the Properties of a Modified Generalized Beverton-Holt Equation in Ecology Models

This paper is devoted to the study of a generalized modified version of the well-known Beverton-Holt equation in ecology. The proposed model describes the population evolution of some species in a certain habitat driven by six parametrical sequences, namely, the intrinsic growth rate (associated with the reproduction capability), the degree of sympathy of the species with the habitat (described by a so-called environment carrying capacity), a penalty term to deal with overpopulation levels, the harvesting (fishing or hunting) regulatory quota, or related to use of pesticides when fighting damaging plagues, and the independent consumption which basically quantifies predation. The independent consumption is considered as a part of a more general additive disturbance which also potentially includes another extra additive disturbance term which might be attributed to net migration from or to the habitat or modeling measuring errors. Both potential contributions are included for generalization purposes in the proposed modified generalized Beverton-Holt equation. The properties of stability and boundedness of the solution sequences, equilibrium points of the stationary model, and the existence of oscillatory solution sequences are investigated. A numerical example for a population of aphids is investigated with the theoretical tools developed in the paper.

A Control Theory point of view on Beverton–Holt equation in population dynamics and some of its generalizations

This paper is devoted to develop some “ad hoc” Control Theory formalism useful for the famous Beverton–Holt equation arising in population dynamics. In particular, the inverse equation is redefined for a finite set of consecutive samples under the equivalent form of a discrete linear dynamic system whose input sequence is defined by the sequence of carrying capacity gains and the unforced dynamics is directly related to the intrinsic growth rate. For that purpose, the environment carrying capacity gains are allowed to be time-varying and designed for control purposes. The controllability property is also investigated on this dynamic extended system as well as the stability, equilibrium points and attractor oscillating trajectories. The properties of the dynamic system associated with the Beverton–Holt inverse equation allow extrapolate in a simple dual way the above properties to the standard Beverton–Holt equation. Some generalizations are given for the case when there are extra parameters in the equation or when the system is subject to the presence of additive disturbances. In all cases, a reference model being also of Beverton–Holt type is proposed to be followed by the control system.