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We present a bioeconomic fishery model with variable price based on supply and demand. The model considers three variables: resource biomass, fishing effort, and market price. Rapid price changes classify the fishery system as a slow–fast system. Two main cases arise: catastrophic equilibrium (CE) with overexploitation and fish extinction, and a sustainable fishery equilibrium. We aim to determine conditions for maximum sustainable yield by optimizing cost per unit of fishing effort. Additionally, we explore conditions to prevent extinction and restore sustainable fishing. Marine protected areas (MPAs) can restore fish stocks by destabilizing the CE and maintaining a stable positive equilibrium. A model incorporating an MPA and fish movement is proposed, suggesting a small carrying capacity-to-surface area ratio in the fishing zone. Optimal total catch can be achieved by selecting the appropriate MPA surface area. [ABSTRACT FROM AUTHOR]

In this paper, we propose to study a fishery model with variable price. We assume that the price evolves much faster than the rest of the system. Under certain assumptions, this makes it possible to consider the fishery system as a slow–fast system, on two time scales, and to study it with a reduced model of dimension two. Two main cases can occur. The first one which we called catastrophic equilibrium corresponds to over-exploitation leading to fish extinction and a booming price. The second case corresponds to a sustainable fishery equilibrium which is stable. The possible effects of the creation of marine protected areas (MPAs), sites where fishing is prohibited, on the fish stock and fishery are evaluated. We show that MPAs can have a positive effect on the restoration of depleted fish stocks by destabilizing the catastrophic equilibrium and keeping only one positive equilibrium which will be globally asymptotically stable. This problem is addressed by proposing a model with MPA for the fish dynamics. Fish are assumed to move between MPA and fishing area and are subject to harvesting through fishing. We show that to avoid the extinction of the stock and stabilize the fishery in the long term, it is necessary to define a fishing zone such that the ratio of its carrying capacity to its surface is small enough. We further show that with a judicious choice of the surface area of the MPA, it is possible to optimize the total capture. [ABSTRACT FROM AUTHOR]

In this paper we consider a general macro-model describing a metapopulation consisting of several interacting with each other subpopulations connected through a network, with the rules of interactions given by a system of ordinary differential equations. For such a model we construct two different micro-models in which each subpopulation has its own structure and dynamics. Precisely, each subpopulation occupies an edge of a graph and its dynamics is driven, respectively, by diffusion or transport along the edge. The interactions between the subpopulations are described by interface conditions at the nodes which the edges are incident to. We prove that with an appropriate scaling, roughly speaking with, respectively, fast diffusion or fast transport combined with slow exchange at the nodes, the solutions of the micro-models can be approximated by the solution to the macro-model. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of the optimal size and number of marine reserves to achieve maximum value in commercial fisheries. A simplified network planning situation is analyzed to optimize the size and number of marine reserves. We consider a general 2L-patch model of harvesting population dynamics with continuous time. Fish movements between the sites, as well as vessel displacements between the fishing sites, are assumed to take place at a faster time scale than the variation of the stock and the change of the fleet size. We take advantage of these two time scales to derive a reduced model governing the dynamics of the total fish stock and the total fishing effort. This reduced model is used to determine the optimal size and number of a marine reserves in order to maximize the catch at equilibrium. We show that the optimal number that maximizes the total fish catch at equilibrium depend intricately on the size of the reserve: A small number of reserves is optimal when the size of the reserve is higher and inverse. [ABSTRACT FROM AUTHOR]

Levels of a complex system are characterized by the fact that they admit a closed functional description in terms of concepts and quantities intrinsic to that level. Several ideas have come up so far in order to make the notion of a closed description precise. In this paper, we present four of these approaches and investigate their mutual relationships. Our study is restricted to the case of discrete dynamical systems, where the different levels are linked by a coarse-graining of variables and states of the system. [ABSTRACT FROM AUTHOR]

We present a classical interspecific competition model. Individuals compete for a resource on a common patch and can go to a refuge. It is assumed that if species would remain on the competition patch, species 1 survives and species 2 would go extinct. Therefore, species 1 is Locally Superior Competitor (LSC) and species 2 Locally Inferior Competitor (LIC). We study the effects of density dependent dispersal from the competition patch to the refuge on the global outcome of competition. We study two cases. The first case considers LSC density dependent dispersal of the LIC trying to escape competition and going to its refuge when the LSC density is large. The second case considers aggressiveness of LIC leading to LIC density dependent dispersal of the LSC. We show that under some conditions, tactic 2 can allow the LIC to survive and even provoke global extinction of the LSC. [ABSTRACT FROM AUTHOR]

A central problem in the study of complex systems is to identify hierarchical and intertwined dynamics. A hierarchical level is defined as an aggregation of the system's variables such that the aggregation induces its own closed dynamics. In this paper, we present an algorithm that finds aggregations of linear dynamical systems, e.g. including Markov chains and diffusion processes on weighted and directed networks. The algorithm utilizes that a valid aggregation with n states correspond to a set of n eigenvectors of the dynamics matrix such that these respect the same permutation symmetry with n orbits. We exemplify the applicability of the algorithm by employing it to identify coarse grained representations of cellular automata. [ABSTRACT FROM AUTHOR]

We present a method for identifying coarse-grained dynamics through aggregation of variables or states in linear dynamical systems. The condition for aggregation is expressed as a permutation symmetry of a set of dual eigenvectors of the matrix that defines the dynamics. The applicability of the condition is illustrated in examples from three different generic classes of reducible Markov chains: systems consisting of independent subsystems, dynamics with symmetries, and nearly decoupled Markov chains. Furthermore we show how the method can be used to coarse-grain cellular automata. [ABSTRACT FROM AUTHOR]