【作者】 张艳；

【导师】 马智慧；

【作者基本信息】 兰州大学 ， 数学·应用数学， 2025， 硕士

【摘要】 随着对捕食系统研究的不断深入,学者们对捕食者-食饵种群间功能反应函数的研究愈发精细.与此同时,自然界中生态系统的不确定性也使得相关捕食系统的研究更加复杂.因此,各类具有现实生态学效应的捕食系统的研究一直受到学者们的广泛关注.然而,随机扰动,时滞效应,生境复杂性效应等相关生态学效应对捕食系统动力学行为影响的研究还不够成熟,相关的动力学理论仍需进一步完善.基于此,本文建立了一类具有生境复杂性效应和广义功能反应的无时滞随机捕食系统,该系统包含两种随机扰动,一种是对系统的线性综合随机扰动,另一种是对捕食者种群攻击率的随机扰动.首先,为了凸显随机扰动对捕食系统动力学行为的影响,针对其对应的确定性系统的平衡点进行了研究,分析了平衡点的存在性以及局部渐近稳定性.其次,运用随机微分方程的存在唯一性定理,Gronwall不等式以及Chebyshev不等式等理论证明了无时滞随机捕食系统全局正解的存在唯一性和各类有界性.同时,结合鞅的性质,强大数定理等理论与方法,对捕食者和食饵种群的平均持久性,灭绝性进行了研究.结果表明,随机扰动对种群动力学行为有着重要影响.当捕食者和食饵种群综合扰动较小时,在捕食者种群攻击率扰动较低,且捕食者种群转化率较高的情况下,捕食者和食饵种群是平均持久的.当捕食者和食饵种群综合扰动适中时,若捕食者种群攻击率扰动较高,只要生境复杂性效应足够强,仍可以确保食饵种群的平均持久,若捕食者种群转化率较低,则捕食者种群将面临灭绝的风险.当捕食者和食饵种群综合扰动较大时,捕食者和食饵种群均会灭绝.最后,本文通过数值模拟对相关理论分析结果进行了验证.基于所构建的无时滞随机捕食系统,首先,本文给出推广后的多时滞随机捕食系统的数学表述,并将原随机捕食系统的线性综合扰动推广为非线性综合扰动.通过构造Lyapunov函数,并利用停时技巧证明了系统全局正解存在唯一性.其次,借助于伊藤公式,Chebyshev不等式等理论与方法验证了系统解的随机有界性和解的二阶矩关于时间的有界性.在此基础上,探讨了捕食者和食饵种群平均持久性,灭绝性的判定条件.结果表明,当捕食者和食饵种群的组合扰动较小时,捕食者和食饵种群均为平均持久的.当捕食者种群攻击率扰动适中时,在特定条件下仍可以保证食饵种群的平均持久性.而当捕食者和食饵种群线性扰动系数较大时,捕食者和食饵种群将面临着灭绝的风险.当捕食者种群线性扰动系数较小时,在捕食者种群转化率较小的情况下,捕食者种群仍将灭绝.再者,通过渐近路径估计和相关定理的证明,发现在长时间尺度上,系统解的增长速率不会超过一次多项式.最后,本文通过数值模拟验证了相关理论分析结果的可靠性.更多还原

【Abstract】 As research on predation systems continues to advance,scholars’exploration of functional responses between predator and prey populations has become increasingly sophisticated.Concurrently,the inherent uncertainties in natural ecosystems have fur-ther complicated these predation systems.Therefore,researches on various predation systems with some realistic ecological effects have always been widely concerned.However,researches on the ecological effects–such as stochastic perturbations,time de-lays,and habitat complexity–on the dynamical behaviors of predation systems remains underdeveloped.The corresponding dynamical theories require further refinement.Based on this,this paper presented a class of the non-delayed stochastic predation system with the effect of habitat complexity and generalized functional response,which includes two types of stochastic perturbations,a linear combined stochastic perturba-tion for the presented system and a stochastic perturbation to the predator population attack rate.Firstly,in order to highlight the impact of stochastic perturbations on the dynamical behaviors of the predation system,the equilibrium point of the correspond-ing deterministic system is considered and the existence and local asymptotic stability of the equilibrium point of the presented system is analyzed.Secondly,the existence and uniqueness of the global positive solution and the boundedness of various classes of the non-delayed stochastic predation system are proved by applying the theorems of existence and uniqueness of stochastic differential equations,Gronwall’s inequality,and Chebyshev’s inequality.Concurrently,the average persistence and extinction of predator and prey populations are investigated by using the properties of martingales and the strong law of large numbers.The results show that stochastic perturbations have a significant impact on the behavior dynamics.If the combined perturbations of predator and prey populations are small,predator and prey populations are on average persistent under low perturbation of predator population attack rate and high predator population conversion rate.If the combined perturbations of predator and prey popula-tions are moderate,the average persistence of prey population could be ensured under high perturbation of attack rate of predator population as long as the effect of habitat complexity is strong enough.At this point,if the predator population conversion rate is low,the predator population will face extinction.If the combined perturbations of predator and prey populations are large,both predator and prey populations will be-come extinct.Finally,the theoretical conclusions being found in this paper are verified by numerical simulations.Based on the non-delayed stochastic predation system,firstly,the mathematical expression of the multi-delayed stochastic predation system was given,and the linear combined perturbations of the corresponding predation system was extended to a non-linear combined perturbations.This paper constructed a Lyapunov function,and the existence and uniqueness of the global positive solution of the system were proved by applying the stopping time technique.Secondly,the It(?)formula,Chebyshev’s inequal-ity were used to verify the stochastic boundedness and the mean boundedness of the second-order moment of solutions of the considered system.Based on this,the av-erage persistence and extinction of the predator and prey populations were discussed.The results show that both the predator and prey populations are on average persistent while the combinatorial perturbation of predator and prey populations is small.The prey population could still be guaranteed to be on average persistent under certain con-ditions while the attack rate perturbation of the predator population is moderate.Both the predator and prey populations will become extinct while the linear perturbation co-efficient of predator and prey populations is large.The predator population will still become extinct under the small conversion rate of the predator population while the coefficient of linear perturbation of the predator population is small.Furthermore,by means of asymptotic path estimation and relevant proof of some theorems,it was found that the growth rate of the system solution would not exceed a first-order polynomial in the long run.Finally,this paper applied numerical simulations to validate the reliability of the theoretical analysis results.更多还原