Ricker Equation Research Articles

«Unfaithful» Pink salmon. Random or regularity?

The reasons of «failed» forecasts for catch of pink salmon at the mainland coast of the Tatar Strait are analyzed. The classical «stock–recruitment» models produce regular errors in the forecasts of runs of pacific salmon under changing conditions of reproduction. To reduce the prediction errors, these models should include parameters of meteorological conditions in the freshwater period of salmon life. In particular case of pink salmon in the Tatar Strait, a relationship is revealed between the year-class strength and precipitation in winter during the period of egg incubation. Including this factor to the model based on Ricker equation increases the proportion of certainty in the variance of recruitment by more than 30 %.

Regularities of reproduction for fall Chum salmon Oncorhynchys keta from the Аmur river

The dependence of the number of progeny of the Amur fall chum salmon Oncorhynchys keta on the number of parents was analyzed. It is shown that the deviations of the actual number of offspring from the theoretical ones, calculated by the Ricker equation, depend on the belonging of the fish to a particular generation. An equation is proposed that describes the dependence of the number of progeny of the autumn chum salmon on the number of parents and reproduction conditions. The problem of the variability of the optimal number of parents due to the change in reproduction levels is discussed.

The effects of evolution on the stability of competing species

Based on evolutionary game theory and Darwinian evolution, we propose and study discrete-time competition models of two species where at least one species has an evolving trait that affects their intra-specific, but not their inter-specific competition coefficients. By using perturbation theory, and the theory of the limiting equations of non-autonomous discrete dynamical systems, we obtain global stability results. Our theoretical results indicate that evolution may promote and/or suppress the stability of the coexistence equilibrium depending on the environment. This relies crucially on the speed of evolution and on how the intra-specific competition coefficient depends on the evolving trait. In general, equilibrium destabilization occurs when , when the speed of evolution is sufficiently slow. In this case, we conclude that evolution selects against complex dynamics. However, when evolution proceeds at a faster pace, destabilization can occur when . In this case, if the competition coefficient is highly sensitive to changes in the trait v, destabilization and complex dynamics occur. Moreover, destabilization may lead to either a period-doubling bifurcation, as in the non-evolutionary Ricker equation, or to a Neimark-Sacker bifurcation.

On the proper periodic Ricker model.

The periodic Ricker equation has been studied by several authors, including the present one. However, the periodic model derived from the original one has not been studied in detail. We show that the model often taken as a periodic Ricker model is a particular case of the original one and compare their dynamics. In particular, we characterize the parameter region where the model has a periodic point of period two, which is globally stable. We also compute the parameter regions where complex behavior is exhibited.

MODELING HARVESTING PROCESSES FOR POPULATIONS WITH NON-OVERLAPPING GENERATIONS

Difference equations are used in order to model the dynamics of populations with non-overlapping generations, since the growth of such populations occurs only at discrete points in time. In the simplest case such equations have the form $N_{t+1}= F(N_t)$, where $N_t >0$ is the population size at a moment of time $t$, and $F$ is a smooth function. Among such equations the discrete logistic equation and Ricker's equation are most often used in practice. In the given paper, these equations are considered width taking into account an effect of harvesting, that is, the equations of the form below are studied $N_{t+1}=r N_t (1- N_t) - c$ and $N_{t+1}= N_t \exp (r(1 - N_t / K )) - c$, where the parameters $r$, $K>0$, $c>0$ are harvesting intensity. Positive equilibrium points and conditions for their stability for these equations were found. These kinds of states are often realized in nature. For practice, periodic solutions are also important, especially with periods $T=2 (N_{t+2} = N_t)$ and $T=3 (N_{t+3} = N_t)$, since, with their existence, by Sharkovskii's theorem, one can do conclusions about the existence of periodic solutions of other periods. For the discrete logistic equation in analytical form, the values that make up the periodic solution with period $T=2$ were found. We used numerical methods in order to find solutions with period $T=3$. For Ricker's model, the question of the existence of periodic solutions can be investigated by computer analysis only. In the paper, a number of computer experiments were conducted in which periodic solutions were found and their stability was studied. For Ricker's model with harvesting, chaotic solutions were also found. As we can see, the study of difference equations gives many unexpected results.

Period-Doubling and Neimark–Sacker Bifurcations of a Beddington Host-Parasitoid Model with a Host Refuge Effect

In this paper, we explore the dynamics of a certain class of Beddington host-parasitoid models, where in each generation a constant portion of hosts is safe from attack by parasitoids, and the Ricker equation governs the host population. Using the intrinsic growth rate of the host population that is not safe from parasitoids as a bifurcation parameter, we prove that the system can either undergo a period-doubling or a Neimark–Sacker bifurcation when the unique interior steady state loses its stability. Then, we apply the new theory to the following well-known cases: May’s model, [Formula: see text]-model, Hassel and Varley (HV)-model, parasitoid-parasitoid (PP) model and [Formula: see text] model. We use numerical simulations to confirm our theoretical results.

Bifurcation analysis and chaos control in a discrete‐time plant quality and larch budmoth interaction model with Ricker equation

We investigate the dynamics of two‐dimensional discrete‐time model of leaf quality and larch budmoth interaction with Ricker equation. More precisely, the qualitative behavior of larch budmoth model is discussed in which the effect of food source upon the moth population is through intrinsic growth rate. We find the parametric conditions for local asymptotic stability of the unique positive fixed point. It is also proved that under certain parametric conditions, the system undergoes period‐doubling bifurcation with the help of center manifold theory. The parametric conditions for existence and direction of Neimark‐Sacker bifurcation at positive fixed point is investigated with the help of standard mathematical techniques of bifurcation theory. The chaos control in the system is discussed through implementation of hybrid control methodology. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long‐term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirms the presence of chaotic behavior in the system.

Bifurcation in the almost periodic Ricker map

ABSTRACTIt was shown in an earlier publication that the Ricker equation with Almost Periodic coefficient with finitely generated frequency module and with average value lying in has Almost Periodic solutions lying on a torus of the same dimension as the (finite) number of independent generators of the frequency module of the coefficient. The hull H of the coefficient is itself a torus homeomorphic to and isomorphic to as an Abelian group.In this paper, we find a bifurcation function and show that if the magnitude of the oscillatory part of the coefficient is increased and the average part is then increased above a bifurcation curve a pair of tori bifurcate, each of which is invariant under the composition of two Ricker maps in sequence and enjoys all the properties as in the pre-bifurcation case. In addition, we show that prior to bifurcation there is a diffeomorphism mapping the torus H onto .It is also surprising to notice that the stability interval is increased as the magnitude of the oscillatory part of the coefficient is increased. Finally the case of a frequency module that is infinitely generated is treated.

Dynamics of the degenerate 2D Ricker equation

The dynamics of the 2D Ricker equation depend on the positioning of its isoclines. We give a complete description for the parameter values where the two isoclines overlap. In this case, we show the plane is covered by a family of invariant curves and that the restriction to each is conjugate to a family of one‐dimensional population maps. Properties of these 1D maps, including global stability, are established, and we discuss the implications for the 2D Ricker equation.

Regularities of reproduction for pink salmon Oncorhynchus gorbuscha from the Iska River

Reasons for rapid rise of pink salmon landings in the Sakhalin Bay (Okhotsk Sea) in the early 21 century are analyzed. Dependence of its juveniles and adults numbers on the number of their parents is well described with Ricker equation, and the recruitment depends linearly on the number of juveniles in the river. Several factors of the pink salmon reproduction are compared between the early 21 century and previous period. There is found that recent in­crease of the catches could not be reasoned by favorable changes of environments, decreasing of juveniles mortality either in the river or in the sea, extending of spawning grounds, growth of spawning stock or number of juveniles. The only possible reason for significant rise of the landings could be an increasing by-catch of transitory stocks.

The spatial dynamics of ecosystem engineers

The changes on abiotic features of ecosystems have rarely been taken into account by population dynamics models, which typically focus on trophic and competitive interactions between species. However, understanding the population dynamics of organisms that must modify their habitats in order to survive, the so-called ecosystem engineers, requires the explicit incorporation of abiotic interactions in the models. Here we study a model of ecosystem engineers that is discrete both in space and time, and where the engineers and their habitats are arranged in patches fixed to the sites of regular lattices. The growth of the engineer population is modeled by Ricker equation with a density-dependent carrying capacity that is given by the number of modified habitats. A diffusive dispersal stage ensures that a fraction of the engineers move from their birth patches to neighboring patches. We find that dispersal influences the metapopulation dynamics only in the case that the local or single-patch dynamics exhibit chaotic behavior. In that case, it can suppress the chaotic behavior and avoid extinctions in the regime of large intrinsic growth rate of the population.

Almost periodicity, Ricker map, Beverton-Holt map and others, a general method

It is known that under the certain conditions on the coefficient the Ricker difference equation (or map) has a fixed point that is globally asymptotically stable with respect to the positive reals. We show here that under the same conditions, the Ricker equation with almost periodic coefficient has a globally asymptotically stable almost periodic solution with the same frequency module as the coefficient. This is accomplished by showing that the omega limit set of an asymptotically stable solution is a covering space of the omega limit set of the coefficients and the flow on is uniquely reversible. We provide a unified framework that allows us to conclude that any system of maps in finite dimensions that has an orbit that is bounded and whose omega limit set is asymptotically stable, also has the property that certain attributes of the coefficients (periodicity, almost periodicity) can be carried over, or lifted to the solution. In particular if the successive compositions are bounded and have Fréchet derivatives with spectrum inside the unit circle in the complex plane then the above conclusions apply.

Global stability in the 2D Ricker equation revisited

We offer two improvements to prior results concerning global stability of the 2D Ricker Equation. We also offer some new methods of approach for the more pathological cases and prove some miscellaneous results including a special nontrivial case in which the mapping is conjugate to the 1D Ricker map along an invariant line and a proof of the non-existence of period-2 points.

Global stability in the 2D Ricker equation

We improve a previous result for the 2D Ricker equation by reducing an infinite number of topological conditions to a finite number. We also give sufficient conditions in terms of the parameters where many of these topological conditions are satisfied. We also discuss the various pathologies that occur for other parameter choices.

Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect

The $\lambda$-Ricker equation has, for certain values of the parameters, an unstable fixed point giving rise to the Allee effect, and an attracting fixed point, the carrying capacity. The $k$-periodic $\lambda$-Ricker equation is studied and parameter intervals are determined for which there exist a $k$-periodic Allee state and a $k$-periodic attracting state.

A conjecture on the stability of the periodic solutions of Ricker’s equation with periodic parameters

In this work, we propose a conjecture about the stability of the periodic solutions of the Ricker equation with periodic parameters, which goes beyond the existing theory, and for the special case of period-two parameters we analytically show the conjecture is true. For this case we show that the stability region in parameter space obtained from the conjecture is larger than a previously proposed stability region. The period-three case is investigated numerically and similar extensions are realized. This suggests that the current theory cited in this paper, while giving sufficient conditions for stability is far from optimal.

A note on periodic Ricker maps

In this note, we study the k-periodic Ricker equation for and establish the following Assume p n ∊ (0, 2). Then the Ricker equation (1.1) has a globally asymptotically stable k-periodic solution ....

Evolutionary Predictions Should Be Based on Individual‐Level Traits

Recent theoretical studies have analyzed the evolution of habitat specialization using either the logistic or the Ricker equation. These studies have implemented evolutionary change directly in population-level parameters such as habitat-specific intrinsic growth rates r or carrying capacities K. This approach is a shortcut to a more detailed analysis where evolutionary change is studied in underlying morphological, physiological, or behavioral traits at the level of the individual that contribute to r or K. Here we describe two pitfalls that can occur when such a shortcut is employed. First, population-level parameters that appear as independent variables in a population dynamical model might not be independent when derived from processes at the individual level. Second, patterns of covariation between individual-level traits are usually not conserved when mapped to the level of demographic parameters. Nonlinear mappings constrain the curvature of trade-offs that can sensibly be assumed at the population level. To illustrate these results, we derive a two-habitat version of the logistic and Ricker equations from individual-level processes and compare the evolutionary dynamics of habitat-specific carrying capacities with those of underlying individual-level traits contributing to the carrying capacities. Finally, we sketch how our viewpoint affects the results of earlier studies.

FOOD AS A MECHANISM OF DENSITY-DEPENDENT REGULATION OF BREEDING NUMBERS IN THE MERLIN,FALCO COLUMBARIUS

Population regulation of Merlins (Falco columbarius) was studied during 16 years in a small area supporting up to 11 breeding pairs, and during seven years in an expanded area where breeding numbers varied between 28 and 38 pairs. During the study period, the previously regular 4-yr cyclicity in rodent abundance disappeared, for unknown reasons. The functional form of density dependence agreed with a version of the Ricker equation, which showed that there was direct density dependence in the number of Merlin pairs attempting to breed. Moreover, in seasons with high small-rodent density, local population density of Merlins increased, and breeding territories were established in areas that, in other years, did not sustain breeders. Overwinter loss of males was directly related to breeding population size in the preceding year, as well as to small-rodent density. In contrast, overwinter loss of females appeared to be independent of breeding population size and food abundance in the preceding year.

Use of coupled oscillator models to understand synchrony and travelling waves in populations of the field vole Microtus agrestis in northern England

1. Earlier studies have reported that field vole Microtus agrestis populations in Kielder Forest, UK, exhibit typical 3–4‐year cyclical dynamics, and that the observed spatiotemporal patterns are consistent with a travelling wave in vole abundance moving along an axis south‐west–north‐east at approximately 19 km year–1. One property of this wave is that nearby populations fluctuate more synchronously than distant ones, with correlations falling lower than the average for the sampling area beyond approximately 13 km. 2. In this paper we present a series of models that investigate the possibility that both the observed degree of synchrony and the travelling wave can be explained as a simple consequence of linking a series of otherwise independently oscillating populations. Our ‘coupled oscillator’ models consider a series of populations, distributed either in a linear array or in a two‐dimensional regular matrix. Local population fluctuations, each with a 3–4‐year period, were generated using either a Ricker equation or a set of discrete‐time Lotka–Volterra equations. Movement among populations was simulated either by a fixed proportion of each population moving locally to their nearest neighbour populations, or the same proportion being distributed via a continuous geometric function (more distant populations receiving less). 3. For a variety of different ways of generating cycles and a number of different movement rules, local exchange between oscillating populations tended to generate synchrony domains that extended over a large number of populations. When the rates of exchange between local populations were relatively low, then permanent travelling waves emerged, especially after an initial invasion phase. There was a non‐linear relationship between the amount of dispersal and the domain of synchrony that this movement generated. Furthermore, the observed spatiotemporal patterns that emerged following an initial invasion phase were found to be highly dependent on the extreme distances reached by rare dispersers. 4. As populations of voles are predominantly distributed in grassland patches created by clear‐cutting of forest stands, we estimated the mean patch diameter and mean interpatch distance using a geographical information system (GIS) of the forest. Our simplified models suggest that if as much as 5–10% of each vole population dispersed a mean of 178 m between clear‐cuts per generation, then this would generate a synchrony domain and speed of wave in the region of 6–24 km (per year), which is reasonably consistent with the observed synchrony domain and speed. Much less dispersal would be capable of generating this scale of domain if some individuals occasionally moved beyond the nearest‐neighbour patch. 5. While we still do not know what causes the local oscillations, our models question the need to invoke additional factors to explain large‐scale synchrony and travelling waves beyond small‐scale dispersal and local density‐dependent feedback. Our work also suggests that the higher degrees of synchrony observed in Fennoscandian habitats compared with Kielder may be due in part to the relative ease of movement of voles in these former habitats. As our work confirms that the rates of exchange among local populations will have a strong influence on synchrony, then we anticipate that the spatiotemporal distribution of clear‐cuts will also have an important influence on the dynamics of predators of voles.