### Proportional threshold harvesting in discrete-time population models

Threshold-based harvesting strategies tend to give high yields while protecting the exploited population. A significant drawback, however, is the possibility of harvesting moratoria with their socio-economic consequences, if the population size falls below the threshold and harvesting is not allowed anymore.

Proportional threshold harvesting PTH is a strategy, where only a fraction of the population surplus above the threshold is harvested.

It has been suggested to overcome the drawbacks of threshold-based strategies. Here, we use discrete-time single-species models and rigorously analyze the impact of PTH on population dynamics and stability.

We find that the population response to PTH can be markedly different depending on the specific population model. Reducing the threshold and allowing more harvest can be destabilizing for the Ricker and Hassell mapstabilizing for the quadratic mapor both for the generalized Beverton—Holt map. Similarly, management actions in the form of increasing the threshold do not always improve population stability—this can also be due to bistability.

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Our results therefore emphasize the importance of a rigorous analysis in investigating the impact of PTH on population stability. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Ecol Lett — Google Scholar. Science — Braverman E, Liz E Global stabilization of periodic orbits using a proportional feedback control with pulses. Nonlinear Dyn — Math Biosci — Clark CW, Kirkwood GP On uncertain renewable resource stocks: optimal harvest policies and the value of stock surveys.

J Environ Econ Manag — Cull P Population models: stability in one dimension. Bull Math Biol — Phys Lett A — Fish Res — Doebeli M Dispersal and dynamics. Theor Popul Biol — J Differ Equ Appl — Enberg K Benefits of threshold strategies and age-selective harvesting in a fluctuating fish stock of Norwegian spring spawning herring Clupea harengus.

Marine Ecol Prog Ser — J Theor Biol — FAO Code of conduct for responsible fisheries. Fieberg J Role of parameter uncertainty in assessing harvest strategies. N Am J Fish Manag — Franco D, Liz E A two-parameter method for chaos control and targeting in one-dimensional maps.Metrics details.

In this paper, a discrete-time analog of a viral infection model with nonlinear incidence and CTL immune response is established by using the Micken non-standard finite difference scheme.

The basic properties on the positivity and boundedness of solutions and the existence of the virus-free, the no-immune, and the infected equilibria are established. By using the Lyapunov functions and linearization methods, the global stability of the equilibria for the model is established.

As is well known, viruses have caused the abundant types of epidemics and are alive almost everywhere on Earth, infecting people, animals, plants, and so on. Therefore, it is important to study viral infection, which can supply theoretical evidence for controlling a disease to break out. In the past years, many authors have studied continuous time viral infection models which are described by the differential equations.

See, for example, [ 1 — 28 ] and the references cited therein. In [ 1 ], Hattaf et al. The dynamic behaviors of the model are studied.

Although model 1 is simple, model 1 is very important in viral epidemiology, which can show ample viral behaviors. The authors studied the local and global stability of the equilibria and the permanence of the model. In [ 26 ], Hattaf and Yousfi proposed the following discrete-time analog directly for model 1 by using the NSFD scheme:. The global asymptotic stability of the disease-free equilibrium and the chronic infection equilibrium is established by constructing the suitable Lyapunov functions.

In [ 28 ], the authors extended model 2 to the delayed case. By using the method of Lyapunov functions, the authors established the global asymptotic stability of the disease-free equilibrium and the chronic infection equilibrium with no restriction on the time-step size. In general, our target is to eliminate and control the virus and infected cells. For all this, many authors have noted that the immune response takes great effect to eliminate and control the virus and infected cells because CTL cytotoxic T lymphocyte cells affect the virus load.

Therefore, a four dimension continuous time virus dynamical model with Beddington-DeAngelis incidence rate and CTL immune response was studied by Wang, Tao and Song in [ 2 ].

The model proposed is as follows:. The authors established the global stability of the disease-free equilibrium, the immune-free equilibrium, and the endemic equilibrium. Motivated by the above works, in this paper we consider a discrete-time analog of a class of continuous time virus dynamical models with nonlinear incidence and CTL immune response which is established by using NSFD scheme.

The model is proposed in the following form:. In this paper, our main purpose is to study the threshold dynamics of model 4. The basic properties on the positivity and boundedness of solutions and the existence of the virus-free equilibrium, the no-immune equilibrium and the infected equilibrium are established. By using the Lyapunov functions and linearization methods, we will establish a series of criteria to ensure the stability of the equilibria for model 4.

Bigfoot trailers for sale by ownerFurthermore, numerical simulations are given. This paper is organized as follows.We hope this content on epidemiology, disease modeling, pandemics and vaccines will help in the rapid fight against this global problem.

Click on title above or here to access this collection. A class of scalar semilinear parabolic equations possessing absorbing sets, a Lyapunov functional, and a global attractor are considered. The dynamical properties of various finite difference and finite element schemes for the equations are analysed. The existence of absorbing sets, bounded independently of the mesh size, is proved for the numerical methods. Discrete Lyapunov functions are constructed to show that, under appropriate conditions on the mesh parameters, numerical orbits approach steady state solutions as discrete time increases.

However, it is shown that insufficient spatial resolution can introduce deceptively smooth spurious steady solutions and cause the stability properties of the true steady solutions to be incorrectly represented. Furthermore, it is also shown that the explicit Euler scheme introduces spurious solutions with period 2 in the timestep. As a result, the absorbing set is destroyed and there is initial data leading to blow up of the scheme, however small the mesh parameters are taken.

To obtain stabilization to a steady state for this scheme, it is necessary to restrict the timestep in terms of the initial data and the space step. Implicit schemes are constructed for which absorbing sets and Lyapunov functions exist under restrictions on the timestep that are independent of initial data and of the space step; both one-step and multistep BDF methods are studied.

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SIAM J. Related Databases. Web of Science You must be logged in with an active subscription to view this. Keywords attractorsabsorbing setsLyapunov functionsspurious solutions. Publication Data. Publisher: Society for Industrial and Applied Mathematics. Elliott and A. Cited by Energy-stable predictor—corrector schemes for the Cahn—Hilliard equation. Journal of Computational and Applied MathematicsIn this paper, a class of discrete SIRS epidemic models with disease courses is studied.

## Global Dynamical Properties of Rational Higher-Order System of Difference Equations

The basic reproduction number R 0 is computed. The main results on the permanence and extinction of the disease are established. In recent years, more and more attention has been paid to the discrete-time epidemic models. There are several reasons for that. Firstly, since the statistic data about a disease is collected by day, week, month or year, it is more direct, more convenient and more accurate to describe the disease by using the discrete-time models than the continuous-time models; secondly, the discrete-time models have more wealthy dynamical behaviors; for example, the single-species discrete-time models have bifurcations, chaos and other more complex dynamical behaviors.

For a discrete-time epidemic model, we see that at the present time, the main research subjects are the computation of the basic reproduction number, the local and global stability of the disease-free equilibrium and endemic equilibrium, the extinction, persistence and permanence of the disease, and the bifurcations, chaos and more complex dynamical behaviors of the model, etc.

Many important and interesting results can be found in articles [ 1 — 24 ] and the references cited therein. In [ 4 ], the next generation matrix approach for calculating the basic reproduction number is summarized for discrete-time epidemic models. As applications, six disease models have been developed for the study of two emerging wildlife diseases: hantavirus in rodents and chytridiomycosis in amphibians.

The comparison of deterministic and stochastic SIS and SIR type epidemic models in discrete time is discussed in [ 3 ]. In [ 89 ], the discrete-time SIS type epidemic models with periodic environment and with disease-induced mortality in density-dependence, respectively, are investigated.

Vzd�l�v�n�, v�daIn [ 11 ], Izzo and Vecchio proposed an implicit nonlinear system of difference equations which represents the discrete counterpart of a large class of continuous models concerning the dynamics of an infection in an organism or in a host population.

They also studied the limiting behavior of the discrete model and derived the basic reproduction number. Izzo, Muroya and Vecchio in [ 10 ] proved the globally asymptotic stability of the disease-free equilibrium for a general discrete-time model of population dynamics in the presence of an infection. For the discrete epidemic model with immigration of infectives, by adopting the means of the nonstandard discretization method from continuous epidemic, Jang and Eiaydi in [ 12 ] studied the globally asymptotic stability of the disease-free equilibrium, the locally asymptotic stability of the endemic equilibrium and the strong persistence of the susceptible class.

Li and Wang in [ 15 ] discussed a SIS type discrete epidemic model with stage structure, where Beverton-Holt type and Richer type recruitment rates were considered, the global stability of the disease-free equilibrium and the dynamical complexity were investigated.

In [ 17 ], the sufficient and necessary conditions for the global stability of the endemic equilibrium were established for a discrete epidemic model for the disease with immunity and latency in a heterogeneous host population.

In [ 19 ], the bifurcations and chaos were proved in a discrete epidemic model with nonlinear incidence rates. In [ 24 ], a discrete mathematical model is formulated to investigate the transmission and control of SARS in China, where the basic reproductive number is obtained as a threshold to determine the asymptotic behavior of the model.

Particularly, in [ 18 ] the authors studied the following class of disease epidemic models with the spread of an infection in a host population:. The global stability of disease-free equilibrium and endemic equilibrium and the permanence of the disease were obtained.

However, we know that many diseases have different disease courses, for example, tuberculosis, syphilis, AIDS, etc. Therefore, taking into account the epidemic models with disease courses is very important since disease pathogen bacteria with different course may have different reproduction and survival capacities, which indirectly influences the population growth.

Under a different disease course, the transmission rate, the mortality and other vital parameters will be different [ 25 — 27 ]. Motivated by the above results, in this paper, we consider a class of discrete-time epidemic models with disease courses. We introduce the following assumptions. Base on the above assumptions, a class of discrete-time epidemic dynamical models with m disease courses can be established as follows:. For model 1we always assume that the following basic hypotheses hold.

The reason for the above arguments is based on two considerations. On the one hand, it is influenced by the works given in [ 1118 ]; on the other hand, for the sake of convenience for mathematical analysis, especially, the positivity of solutions in model 1.

In this paper, by developing the methods given in [ 101118 ], we will give the explicit expression of the basic reproduction number R 0. The criteria on the permanence and extinction of the disease will be established.

This paper is organized as follows.Skiba, A K, Swan, Winford H. Full references including those not matched with items on IDEAS More about this item Statistics Access and download statistics Corrections All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:mcr:wpdief:wpaper See general information about how to correct material in RePEc.

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Please note that corrections may take a couple of weeks to filter through the various RePEc services. Economic literature: papersarticlessoftwarechaptersbooks. FRED data. We study the dynamics shown by the discrete time neoclassical one-sector growth model with differential savings as in Bohm and Kaas while assuming a nonconcave production function in the form given by Capasso et. We prove that complex features are exhibited related both to the structure of the coexixting attractors and to their basins.

**Lecture 1, Introduction - MIT RES.6.007 Signals and Systems, Spring 2011**

We also show that complexity emerges if the elasticity of substitution between production factors is low enough and shareholders save more than workers, confirming the results obtained while considering concave production functions see, for instance, Brianzoni et al. Handle: RePEc:mcr:wpdief:wpaper as. More about this item Statistics Access and download statistics. Corrections All material on this site has been provided by the respective publishers and authors.

Louis Fed.We study the dynamics shown by the discrete time neoclassical one-sector growth model with differential savings while assuming a nonconcave production function. We prove that complex features exhibited are related both to the structure of the coexixting attractors and to their basins.

Billing document table in sapWe also show that complexity emerges if the elasticity of substitution between production factors is low enough and shareholders save more than workers, confirming the results obtained while considering concave production functions. The standard one-sector Solow-Swan model see [ 12 ] represents one of the most used frameworks to describe endogenous economic growth. It describes the dynamics of the growth process and the long-run evolution of the economic system.

Let be the production function in intensive form, mapping capital per worker into output per workerthen the Solow-Swan growth model describing the evolution of the state variable in a discrete time setup is given by where is the constant labor force growth rate, is the depreciation rate of capital, and is the constant saving rate.

Most papers on economic growth considering the Solow-Swan or neoclassical model used the Cobb-Douglas specification of the production function, which describes a process with a constant elasticity of substitution between production factors equal to one. It is quite immediate to observe that in this formulation the system monotonically converges to the steady state i.

More recently, several contributions in the literature have considered the Constant Elasticity of Substitution CES production function, in order to study growth models with elasticity of substitution that can be either greater or lower than one see for instance [ 34 ]. In fact, as underlined in Klump and de La Grandville [ 5 ], the elasticity of substitution between production factors plays a crucial role in the theory of economic growth since it represents one of the determinants of the economic growth level.

Anyway the long run dynamics is still simple. Another consideration is that the standard one-sector growth model does not take into account that different groups of agents workers and shareholders have constant but different saving propensities. Such an issue has been studied by many authors i. In fact different but constant saving propensities make the aggregate saving propensity nonconstant and dependent on income distribution, so that multiple and unstable equilibria can occur.

They use a generic production function satisfying the weak Inada conditions, that is. The authors show that instability and topological chaos can be generated in this kind of model. Brianzoni et al. The authors proved that multiple equilibria are likely to emerge and that complex dynamics can be exhibited if the elasticity of substitution between production factors is sufficiently low.

The results obtained prove that production function elasticity of substitution plays a central role in the creation and propagation of complicated dynamics in growth models with differential saving. As a further step in this field, Brianzoni et al. The use of the VES production function allows to take into account that the elasticity of substitution between production factors is influenced by the level of economic development. The authors prove that the model can exhibit unbounded endogenous growth when the elasticity of substitution between labour and capital is greater than one, as it is quite natural while the variable elasticity of substitution is assumed and differently from CES confirming the results obtained by Karagiannis et al.

Furthermore, the results obtained aim at confirming that the production function elasticity of substitution is responsible for the creation and propagation of complicated dynamics, as in models with explicitly dynamic optimizing behavior by the private agents see Becker [ 17 ] for a survey about these models. For many economic growth models based on intertemporal allocation, the hypothesis of a concave production function has played a crucial role.

In fact the production function is the most important part of a growth model as it specifies the maximum output for all possible combinations of input factors and therefore determines the way the economic model evolves in time. Let us focus on the meaning of condition from an economic point of view.

Diagram of tecno w3We take into account a region with almost no physical capital, that is there are no machines to produce goods, no infrastructure, and so forth. Then the previous condition states that it is possible to gain infinitely high returns by investing only a small amount of money.

This obviously cannot be realistic since before getting returns it is necessary to create prerequisites by investing a certain amount of money. After establishing a basic structure for production, one might still get only small returns until reaching a threshold where the returns increase greatly to the point where the law of diminishing returns takes effect.

In literature this fact is known as poverty traps.In this paper, global dynamical properties of rational higher-order system are explored in the interior of.

Fantasy books where the bad guy winsIt is explored that under certain parametric conditions, the discrete-time system has at most eight equilibria. By the method of linearization, local dynamics has been explored.

It is explored that positive solution of the system is bounded, and moreover fixed point is globally stable if. It is also investigated that the positive solution of the system under consideration converges to.

### Global dynamics in a class of discrete-time epidemic models with disease courses

Lastly, theoretical results are confirmed by numerical simulation. The presented work is significantly extended and improves current results in the literature. It is a well-known fact that difference equations arise naturally as discrete analogues and as numerical solutions of differential as well as delay differential equations having applications in many fields like physics, biology, economy, and ecology.

Recently, a lot of studies have been conducted concerning the global dynamics of difference equations and their systems [ 3 — 19 ]. It is really not easy to understand global dynamics of difference equations along their systems; particularly, investigating the global behavior of higher-order equations is a challenging job in recent years.

Therefore, investigating the global dynamics of such difference equations along their systems is worth further consideration.

For illustration, Gibbons et al. Shojaei et al. Bajo and Liz [ 23 ] investigated the dynamics of the following difference equation: where and are positive constants. Zhang et al. Recently, Qureshi and Khan [ 25 ] investigated the global dynamics of the following rational system, which is extension of the work [ 21 — 24 ]: where and are positive constants.

Inspired from aforesaid studies, we will extend the work studied by numerous authors [ 21 — 25 ] to investigate global dynamics of the following - dimensional system: where and are positive constants. The organization of this paper is as follows.

In Section 2existence of equilibria in and corresponding linearized form are investigated. Section 3 deals with the study of local dynamics about equilibrium points.

Boundedness of positive solution for the discrete-time system is studied in Section 4. Further, global dynamics about is explored in Section 5. In Section 6we studied the rate of convergence which converges to of the system. Theoretical results are numerically verified in Section 7while concluding remarks are given in Section 8.

The existence of equilibrium solution in the interior of and linearized form about of system 7 are investigated in this section. So, existence of equilibrium solution can be summarized as the following lemma. Lemma 1. In the interior ofdiscrete-time system 7 has at most eight equilibria. More precisely, iis the unique boundary point of discrete-time system 7.

Hereafter, we establish the corresponding linearized form of 7.

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