A NEW FLEXIBLE DISCRETE-TIME MODEL FOR STABLE POPULATIONS

. We propose a new discrete dynamical system which provides a ﬂexible model to ﬁt population data. For diﬀerent values of the three involved parameters, it can represent both globally persistent populations (compen-satory or overcompensatory), and populations with Allee eﬀects. In the most relevant cases of parameter values, there is a stable positive equilibrium, which is globally asymptotically stable in the persistent case. We study how popu- lation abundance depends on the parameters, and identify extinction windows between two saddle-node bifurcations.


1.
Introduction. Discrete-time single-species models are the most appropriate mathematical description of life histories of organisms whose reproduction occurs only once a year during a very short season, and it is assumed that, in the rest of the year, the population is only subjected to mortality, but not to births (see, e.g., [17]). Thus, the between-year dynamics is governed by a first-order difference equation x n+1 = f (x n ), where x n denotes the population at the n-th generation, censused after reproduction. These models are widely used in fisheries, but are well-suited for many other organisms [10,Chapter 4].
The production function f is usually density-dependent (which means that f (x)/x is not constant), and the strength of density dependence is determined by several parameters related to the growth rate (per capita number of offspring), the probability of surviving the reproductive season, the carrying capacity of the environment, and intraspecific cooperation or competition factors. In fisheries, discrete-time models have a long tradition, and the map f is usually referred to as the stock-recruitment relationship. Finding a good stock-recruitment curve for fitting the population dynamics of a particular species has been one of the major problems in theoretical studies of fisheries [3,9,12].
There are different forms of density-dependence, which correspond to different characteristics of the production function f . Density dependence is compensatory if survival decreases with increasing population abundance. In mathematical terms, it is described by conditions f (x) > 0, F (x) < 0, being F (x) = f (x)/x the per capita production function. Perhaps the most well-known compensatory function is the x n+1 = βx γ n 1 + δx n := f (x n ), (1.1) where β, γ, δ are positive parameters. Clearly, equation (1.1) can be obtained as a generalization of the Beverton-Holt (γ = 1) and the Cushing (δ = 0) models. As we show below, (1.1) represents a compensatory population if γ = 1 (actually, it is the Beverton-Holt model), overcompensatory if γ < 1, and depensatory if γ > 1. In the latter case, it usually has a stable positive equilibrium if 1 < γ < 2, and no stable positive equilibria if γ ≥ 2 (displaying a dynamics similar to the Cushing model with γ > 1). Thus, the new model is very flexible. However, it is worth emphasizing that it does not exhibit the complicated behavior typical of such overcompensatory models as the quadratic and the Ricker maps [11]. Equation (1.1) is suitable to model stable populations, that is, populations that tend to an equilibrium in the long-term. For γ > 1, equation (1.1) can be introduced in a phenomenological way multiplying the classical Beverton-Holt model by a density-dependent factor I(x) = x γ−1 . Thus, it can be seen as a model with a factor of positive density dependence I(x) and a factor of negative density-dependence 1/(1 + δx). A similar approach has been used by Avilés [1] to obtain the gamma model x n+1 = βx γ n e −δxn from the Ricker map; see [14] for related models with Allee effects, and [8] for a detailed study of the gamma model. When positive and negative density-dependent factors interact in a population model, an interesting question is whether population abundance increases in response to an increment of parameter γ in the factor of positive density dependence (Avilés refers to γ as the cooperation parameter). We show that this is not always the case, and establish the exact conditions on the involved parameters that ensure that the positive equilibrium is an increasing function of γ.
2. Preliminary results. In our first result, we list several basic properties of the map f : [0, ∞) → [0, ∞) that defines the right-hand side of (1.1).
Proof. The proof is elementary. We just state the simplified expressions for the first and the second derivatives of f for future reference: In Figure 1, we plot different graphs of f , depending on the involved parameters. The case γ = 1 is the so-called Beverton-Holt model [2]. The dynamics for this particular case are well known, and we just state them without proof in the following result: • If β > 1, then lim n→∞ f n (x) = p, for all x > 0.
The case γ ≥ 2 is also elementary because f is increasing and convex. In the next result, we state the dynamics for (1.1) in this case: Proposition 3. If γ > 2, or γ = 2 and β > δ, then equation (1.1) has a unique positive equilibrium p, which is a repeller. Moreover, If γ = 2 and β ≤ δ, then the unique equilibrium of (1.1) is 0, and it is a global attractor.
Actually, the case γ > 2 seems of limited application in population modeling because f (x) increases with superlinear growth as x tends to infinity. The same limitation has been pointed out for the Cushing model x n+1 = βx γ n , with γ > 1 (see [12,Section 3.1.3]). For γ = 2, f increases linearly at infinity; in this particular case, model (1.1) has been derived in a mechanistic way in the context of Allee effects due to mate limitation [6].
In the following sections, we address the cases γ < 1 and 1 < γ < 2. As we will see, equation (1.1) has a unique positive fixed point if γ < 1, and can have 0, 1, or 2 positive fixed points if 1 < γ < 2 (see Figure 1). When f has two fixed points, the smallest one is unstable, and it is usually referred to as the Allee threshold [4]. If population abundance falls below the Allee threshold, then extinction occurs.
3. The overcompensatory case γ < 1. In this section, we prove that equation (1.1) has a globally attracting positive equilibrium if 0 < γ < 1. For it, we shall use the following result from [17, Section 9.3]: ) is continuous, has a unique fixed point p, and is bounded on (0, p). Moreover, assume there exist It is clear from Proposition 1 that f (x) = βx γ /(1+δx) is continuous and bounded if γ < 1. Next we prove that there is a unique positive equilibrium, and it is a global attractor in (0, ∞).

Proof.
A point x > 0 is an equilibrium of (1.1) if and only if it solves equation On the one hand, since γ < 1, We prove that p is asymptotically stable. Indeed, using (2.1) and (3.1), we get from where it is clear that |f (p)| < 1, and therefore p is asymptotically stable. To prove that p is a global attractor, we show that equation f 2 (x) = x has no positive solutions different from p (so Proposition 4 applies). We have: .
We have to demonstrate that equation f (x) = g(x) has a unique positive solution (which is clearly the fixed point p of f ). First, we prove that f is concave in (0, c), where c = γ/(δ(1−γ)) is the unique critical point of f . We know from Proposition 1 that f has a unique inflexion point It is easy to prove that f is concave in the interval (0, d), and c < d. Therefore, f is concave in (0, c).
To conclude the proof, it is enough to show that g is increasing and convex in (0, ∞). Assume this claim is true; then, since f (0 + ) = ∞ and g (0) = 0, equation f (x) = g(x) has at least one positive solution. Let p 1 be the first intersection point of the graphs of f and g. Notice that g (p 1 ) > f (p 1 ), g is increasing and positive in (p 1 , ∞), and f is decreasing in (p 1 , c) and negative in (c, ∞). Thus g (x) > f (x) for all x > p 1 , which ensures that additional intersections are not possible. g is increasing: Indeed, we have: β .
• If β > β * , then p and 0 are asymptotically stable, and A is unstable. Moreover, the basin of attraction of p is (A, ∞), and the basin of attraction of 0 is (0, A). • If β = β * , then p is semistable, with basin of attraction [p, ∞), and 0 is asymptotically stable, with basin of attraction (0, p). • If β < β * , then all solutions of (1.1) converge to 0.
Proof. We recall that a positive equilibrium of (1.1) is a solution of equation f 1 (x) = f 2 (x), where f 1 (x) = βx γ−1 , and f 2 (x) = 1 + δx. Since 1 < γ < 2, f 1 is increasing and concave, and therefore (3.1) can have 0, 1, or 2 positive solutions. It has exactly one positive solution when the graphs of f 1 and f 2 have a tangency at a point p > 0, that is, when the system of equations f 1 (x) = f 2 (x), f 1 (x) = f 2 (x) has a positive solution. This system leads to and therefore the values of x and β that solve the system are (4.1) A simple graphical analysis shows that (3.1) has one solution for β = β * , two solutions for β > β * , and no solution for β < β * .
Next we show that, for β > β * the smallest positive fixed point A is unstable and the largest one p is asymptotically stable.
Since A < p * < p, where p * is defined in (4.1), we get where we have used that the map x → x/(1 + δx) is increasing in (0, ∞).
The basins of attraction are obtained as a simple consequence of the fact that f is increasing.

The influence of γ on population abundance. Extinction windows.
In this section, we study the response of population abundance to a variation of γ ∈ (0, 2). We are only interested in the stable positive equilibrium, which we denote p(γ) to emphasize that we use γ as the bifurcation parameter.
• The values of γ 1 and γ 2 can be calculated using the equation that defines β * in (4.1).
6. Discussion. We have introduced the discrete-time single-species model (1.1), which generalizes the Beverton-Holt and the Cushing models. The main feature of (1.1) is its flexibility, being capable of exhibiting compensatory, overcompensatory, and depensatory growth. For γ < 1, the stock-recruitment curve is unimodal, with population size, x  > 1), or to the extinction equilibrium (β ≤ 1). If γ > 1, then the extinction equilibrium is asymptotically stable, inducing an Allee effect. For γ ∈ (1, 2), the dynamics of (1.1) falls into one of two generic possibilities: extinction (if 0 is the unique equilibrium), or bistability, when the extinction equilibrium coexists with an asymptotically stable positive equilibrium. In the latter case, initial population densities below the Allee threshold A are doomed to extinction, but populations above A persist and converge to the largest positive equilibrium. If γ ≥ 2, then populations above the positive equilibrium grow indefinitely, as in the Cushing model. Although the dynamics of (1.1) is simple, it is still an interesting equation from a dynamical point of view, because it exhibits all types of generic bifurcations of fixed points with an eigenvalue of 1 (saddle-node, transcritical, pitchfork). An interesting feature of the bifurcation diagrams is the possibility of extinction windows when γ is used as the bifurcation parameter.
For γ < 2, the interplay between the three involved parameters determines in a subtle way when the cooperation parameter γ is beneficial for population abundance. While this is true for β > 1 + δ, if β < 1 + δ then increasing γ can have negative effects, leading the population smoothly to extinction ( Figure 5), or even to a sudden collapse (Figure 4 (c), (d)).
The ecological significance of our results about the influence of γ on population abundance is related to cooperation-mediated persistence, and there are some interesting analogies with predator-prey models where predators cooperate to exploit their preys [7,16]. In the one-dimensional model considered here, the population cooperates and exploits resources, for which they compete intraspecifically. We showed that cooperation can help persistence; for example, Figures 4 (c) and 5 (b) show a "rescue" effect: populations that would go extinct in the absence of cooperation can survive for sufficiently large cooperation, provided initial population size is large enough. However, if productivity or competition are too large, cooperation cannot change the outcome (Figures 4 (a), 4 (b) and 5 (a)). This point has been made for predators subject to high disease mortality [7]. Figures 4 (c) and 4 (d) show a somehow counterintuitive effect: population abundance can decrease with increasing cooperation parameter values γ > 1. A possible explanation is that population tends to grow with γ but intraspecific competition for resources becomes stronger because of increasing population size. In these cases, the high value of δ results in a prevalence of competition over cooperation. A similar phenomenon has been observed in a predator-prey model with hunting cooperation [16].