Simple Fishery and Marine Reserve Models to Study the Sloss Problem

Habitat fragmentation is generally believed to be detrimental to the persistence of natural populations. In nature management one therefore tends to prefer many small nature reserves over one single large having equal total area. The paper presents two tractable analytical models to examine whether this preference is warranted in a metapopulation framework with reserves (patches) by formulating the dependence of the density as well as the global catch on the number of reserves. Studying these models, we seek to compare the two different strategies: whether a single large reserve will (1) conserve more species and (2) maintains high yield in fisheries than several small or vice versa? Our results indicate that it is favourable to make several small reserves instead one single large reserve. Indeed, at equilibrium, the density as well as the global catch are bigger in the Several Small (SS) strategy than in the Single Large (SL) one. In addition, in the case of SS strategy, an intermediate reserve number exists which is optimal for catch at equilibrium. The results in this paper may provide important implications for nature conservation. Résumé. La fragmentation de l’habitat est généralement considéré comme préjudiciable à la persistance des populations naturelles. Dans la gestion des milieux naturels, il y a donc tendance à préférer plusieurs petites réserves à une seule ayant la même superficie. Ce papier présente deux modèles mathématiques étudiant ce problème dans le cadre d’une métapopulation avec des zones protégées représentés par des sites discrets. Nous étudions notamment les captures globales en fonction du nombre de réserves. Nous cherchons par cette étude à comparer deux stratégies de préservation : Une seule grande réserve ou plusieurs petites. Nos résultats indiquent qu’il est préférable de mettre plusieurs petites réserves plutôt qu’une seule réserve. En effet, à l’équilibre, la densité ainsi que la capture globale sont plus grandes dans le cas de la stratégie avec plusieurs petites réserves (SS) que dans le cas d’une seule grande (SL). En outre, dans le cas de la stratégie SS, nous démontrons qu’il existe un nombre optimal de reserves qui maximise la capture totale à l’équilibre. Les résultats présentés dans cette contribution sont susceptibles d’avoir des implications importantes dans le domaine de la conservation des espèces. Introduction Marine reserves, or no-take zones, have been recently promoted as a means of managing marine populations for two different goals: preserving biodiversity [1], and managing fisheries [19] to produce the highest yields. Controversy on how to plan marine reserves lead scientists to start the debate on whether single large or several small reserves would be better to conserve biodiversity [16,17], this is known in the ecological literature as the Single Large Or Several Small (SLOSS) problem. The point of view that a single large reserve is 1 Department of Mathematics, Faculty of Sciences, University of Tlemcen, 13000, Algeria. 2 UMI IRD 209 UMMISCO, Centre IRD de lIle de France, 32 Avenue Henri Varagnat, 93143 Bondy Cedex, France. UPMC, Sorbonne University Pierre et Marie Curie-Paris 6, France. c © EDP Sciences, SMAI 2015 Article published online by EDP Sciences and available at http://www.esaim-proc.org or http://dx.doi.org/10.1051/proc/201549007 ESAIM: PROCEEDINGS AND SURVEYS 79 the best and most intuitive strategy for long-term population persistence is mainly based on species area relationships. However, the concept of ”keeping all eggs in the same basket” makes the single large argument weaker, considering the risks involved in allocating all investments in one single area [20]. By the other side, several small areas make populations easy targets for demographic and environmental stochasticity, and effects of genetic degeneration (i.e. inbreeding and bottleneck effects). However, it is also expected that by maintaining several areas, higher habitat variation and, as consequence, higher species richness are achieved at landscape scale. Since the beginning of the SLOSS debate in the middle of 1970s, researchers are testing hypotheses considering different landscape scenarios and distinct taxonomic groups in an attempt to provide a final answer to such dichotomy. Although a simple model cannot provide a definitive answer to this question, it can be used to explore several possibilities. Our goal in this paper is to look in a quantitative this question within the context of two simple deterministic models. First of all, we study the dynamics of a model in the case of no reserve. We then generalize to the case of marine reserve in section 3. The model includes two time scales, a fast one associated to quick movements of fish between sites in comparison to a slow one corresponding to the growth of the fish population and the change of the fleet size. We take advantage of these two time scales to build a reduced model, by applying aggregation methods of variables [6, 9]. The reduced model, called aggregated model, describes the dynamics of the total fish stock and the fishing effort. Studying this aggregated model, we investigate the practical effects of the protection zone on the conservation of population resources and catch at equilibrium. In section 4, we propose a dynamic one-dimensional model of Marine Protected Areas (MPAs) along a coastline to investigate the practical effects of several small reserve on the conservation of population resources and catch at equilibrium. In section 5, we then compare results obtained, particularly using numerical simulations. Our results indicate that having multiple small MPAs may be preferable to having a single large one of the same area. 1. Classical fishery model In this section we summarize some crucial properties of classical fishery models without marine reserves. This will serve as a convenient benchmark in evaluating the impact of marine reserves in later sections. Consider a fishing area of constant size K inhabited by a stock of fish. Let the stock of fish at each point of time be fully described by its biomass, n(t) where t refers to time and E(t) be the fishing effort, at time t. We assume that the population evolves according to the logistic law of growth. The following system describes the time evolution of the fishery [26]:  dn dt = rn(1− n K )− qnE dE dt = (−c+ pqn)E (1) where r is the growth rate of the resource. It is also assumed that the catch is proportional to the fish density and to the fishing effort. q > 0 is a catchability parameter, c represents the cost per unit of fishing effort, and p is the price per unit of fish. This system is well know because it is equivalent to a Lotka-Volterra predator-prey model with logistic growth for prey. It has three equilibria, (0, 0), (K, 0), which is a “fishery-free” equilibrium (FFE), and a unique and possibly positive fishery equilibrium (FE) (n∗, E∗) with  n∗ = c pq E∗ = rq ( 1− c pqK ) (2) 80 ESAIM: PROCEEDINGS AND SURVEYS This equilibrium is globally asymptotically stable when pqK > c, in this case, the fishery is viable and the catch per unit of time at equilibrium is Y ∗ = qn∗E∗ = rc pq ( 1− c pqK ) (3) Keeping all parameters constant except the catchability parameter q, one sees from equation (3) that Y ∗ is a function of q. Y ∗ is positive when q ∈ [ c pK ,∞ [ and reaches a maximum value for q = 2c pK . for this value of q, the optimal catch at equilibrium is rK 4 . 2. Single large MPA We now turn our attention to marine reserves. We begin by considering the impact on the previous classical fishery model. We assume that fishing is prohibited in a certain part of the coast. Let s be the fraction of the protected zone of the total coast. Clearly, being a fraction s ∈ [0, 1]. s = 0 there is no marine reserve and s = 1 indicates that the whole fishing area is a reserve. Then, we assume that this area splits in two sub-zones, with capacity respectively equal to k1 = sK and k2 = (1 − s)K, we denote the stock of the corresponding sub-populations by n1 and n2. These two populations follow two independent evolution laws and this is the reason why we use the ”patch concept”. From now on, we decide that the first patch with capacity k1 is a protected zone where no catch is allowed, whereas in the second patch fishing is allowed (Figure 1). We now Figure 1. Schematic representation of a network of marine reserve with reserve in white and fished area in gray. assume that some exchange of fishes exists between these two patches and that this can occur at a fast time scale τ , whereas fish growth and the dynamics of the fishery occur at a slow time scale t = τ with << 1 being a small dimensionless parameter. It makes sense to assume that fish movement rates are inversely proportional to that carrying capacity of the site they leave [4, 10,22,23]: