Bioeconomic Dynamic Model Of Two Patches For Those Population Densities Obeying The Law Of Gompertz Growth

Population ecology is a branch of ecology in which mathematics is widely used. Itsresearch contains the study of dynamical properties and structures for a given population andthe evolution law of the interaction between the given population and its related populations.Mathematical models provide us with a more intuitive understanding of dynamical behaviorof dynamic interaction between and within patches of population. Through the study of somesimple models, we can obtain practical measurements and suggestions on the speciesprotection and capture issue.In order to provide theoretical basis for the development and protection of biologicalresources, we propose two population models of two patches in which the population densityobey the Gompertz law of growth based on the review and sum-up of previous researchresults, then we analyze the two models with the method of dynamic system and study thefishing effort based on the optimal control theory. The main results of this paper are asfollows:1. A detailed comparison of the rule of the Logistic growth and Gompertz growth isconducted.2.We propose a two-patch model in which the population dendity obey the Gompertz lawof growth,one of the patches is a protected area, the migration going from protected area tothe zone of open-access is nonlinear. First, we prove the system exists an unique nontrivialsingular point and the nontrivial singular point is locally asymptotically stable. We prove thesystem does not exist limit cycle before we prove the global asymptotic stability of thenontrivial singular point. This shows that the population density in the two patches caneventually achieve their equilibrium. Then, we study the bioeconomic equilibrium of thesystem by introducing the factors of price and cost. In the situation of bioeconomicequilibrium, the density of population y varies directly with the cost of capture and inverselywith population price p, the fishing effort E varies directly with the migration rate m andinversely with the cost of capture c. Finally, we study the optimal capture strategy and obtain the economic profit will decreases as the discount rate increases. We present thesingular control path, through the path we can get the optimal equilibrium quantity of theoptimal capture strategy.3. On the basis of the previously considered case, we propose a three-dimensional systemby introducing predator population in the open-access area. First, we prove the system existstwo nontrivial singular pointsP1andP2, and the two nontrivial singular points are locallyasymptotically stable, the nontrivial singular pointsP1degenerate into the nontrivial singularpoint of the system in chapter three, thusP1is globally asymptotically stable. Under thegiven conditions, the global asymptotic stability of the nontrivial singular pointP2is proved.This illustrates that the density of populations will tend to their stable states over time. Weprove population x is permanent. Then, we study the bioeconomic equilibrium of the system,and obtain the bioeconomic equilibrium surface, and investigate the effects of populationdensities when some parameters change. Finally, we study the optimal capture strategy, andobtain the cost of per unit effort equal to the discount value of the future marginal revenue ofinvested effort under the steady state level, and get the optimal capture path. Through thenumerical simulation of example6and7, we find that the numerical solution is in line withthe principle of Volterra.4. Most of the theoretical analysis results of our study are numerically verified by usingthe Mathematica software.