Critical Transitions for Asymptotically Concave or d-Concave Nonautonomous Differential Equations with Applications in Ecology

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
Jesús Dueñas, Carmen Núñez, Rafael Obaya
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引用次数: 0

Abstract

The occurrence of tracking or tipping situations for a transition equation \(x'=f(t,x,\Gamma (t,x))\) with asymptotic limits \(x'=f(t,x,\Gamma _\pm (t,x))\) is analyzed. The approaching condition is just \(\lim _{t\rightarrow \pm \infty }(\Gamma (t,x)-\Gamma _\pm (t,x))=0\) uniformly on compact real sets, and so there is no restriction to the dependence on time of the asymptotic equations. The hypotheses assume concavity in x either of the maps \(x\mapsto f(t,x,\Gamma _\pm (t,x))\) or of their derivatives with respect to the state variable (d-concavity), but not of \(x\mapsto f(t,x,\Gamma (t,x))\) nor of its derivative. The analysis provides a powerful tool to analyze the occurrence of critical transitions for one-parametric families \(x'=f(t,x,\Gamma ^c(t,x))\). The new approach significatively widens the field of application of the results, since the evolution law of the transition equation can be essentially different from those of the limit equations. Among these applications, some scalar population dynamics models subject to nontrivial predation and migration patterns are analyzed, both theoretically and numerically. Some key points in the proofs are: to understand the transition equation as part of an orbit in its hull which approaches the -limit and Abstract Image-limit sets; to observe that these sets concentrate all the ergodic measures; and to prove that in order to describe the dynamical possibilities of the equation it is sufficient that the concavity or d-concavity conditions hold for a complete measure subset of the equations of the hull.

Abstract Image

渐近凹或 d-Concave 非自治微分方程的临界转换及其在生态学中的应用
分析了具有渐近极限 \(x'=f(t,x,\Gamma _\pm (t,x)) 的过渡方程 \(x'=f(t,x,\Gamma _\pm (t,x)) 的跟踪或临界情况的发生。逼近条件只是 \(lim _{t\rightarrow \pm \infty }(\Gamma (t,x)-\Gamma _\pm (t,x))=0/)均匀地在紧凑实集上,因此对渐近方程对时间的依赖没有限制。假设假定映射(x/mapsto f(t,x,\Gamma _\pm (t,x))或它们关于状态变量的导数(d-凹性)在x上是凹性的,但不是映射(x/mapsto f(t,x,\Gamma (t,x))或它的导数的凹性。该分析为分析一参数族 \(x'=f(t,x,\Gamma ^c(t,x))\) 临界转换的发生提供了强有力的工具。新方法大大拓宽了结果的应用领域,因为过渡方程的演化规律可能与极限方程的演化规律有本质区别。在这些应用中,我们从理论和数值两方面分析了一些受非琐碎捕食和迁移模式影响的标量种群动力学模型。证明中的一些关键点是:将过渡方程理解为其全壳中接近-极限集和-极限集的轨道的一部分;观察到这些集集中了所有的遍历度量;证明为了描述方程的动力学可能性,全壳方程的完整度量子集的凹性或d-凹性条件成立就足够了。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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