具适应性的人口疏散模型的整体解应用数学论文

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具适应性的人口疏散模型的整体解

摘要

在人口疏散中，采取怎样的方式疏散(扩散)人口更为有效是一个非常重要的

问题。具适应性表示人口向着资源密集的地方移动(迁移)。这里的疏散不仅仅指人口疏散，还可以表示生物种群的扩散演化。对这些问题的研究在生物学、社会学上有着广泛的应用。

本文考虑了两种完全相同的种群在相同的环境下采取不同的策略——一种

采取随机自由扩散策略，另一种采取具适应性的扩散策略——竞争演化模型，并证明其在整个时间区间上的古典解的存在唯一性。

关键词：扩散，偏微分方程，适应性，整体解存在性

Global Existence for Population Evacuation Model

with Adaptability

Abstract

In the evacuation of the population, ) population more effectively is a very

important question. Adaptive is defined as follows: population move(migration) toward the place where resource is intensive. The evacuation of population, also can say the diffusion of species evolution. Research on these problems is widely used in biology, sociology.

In this paper we consider two identical population that take different

strategies in the same environment—one taking a random free diffusion strategy, another taking diffusion strategy with adaptability –competition evolution model, and prove that the time interval of the classical solution existence uniqueness. Key words: diffusion, partial differential equations, adaptability, existence of global solution

目 录 1. 引言

1.1 研究背景

大多数生物种群的一个明显特征是：它们有着空间分布。所以很自然的要问，种群的扩散过程是如何导致空间分布的模式的，什么样的模式产生怎样的过程，以及为什么生物可以进化到这样的扩散方式。在这方面已经作出了相当大的努力，利用空间模型来解决这些问题。在本文中，我们将探讨空间明确的种群模型,该模型与一个特定的模式，理想自由分布有关。在最初的形式中，理想自由分布是简单的描述生物怎样定位他们自己，如果他们能自由的移动到最合适他们的适应性。一个版本的理想自由分布连续空间可以源自于一种平流分布方程，

该方程基于假定生物向上层局部适应性梯度移动，并且这个适应性随着空间变化、随着拥挤现象下降。我们考虑一个在这个模型中同样包括随机扩散的分布部分的变化。我们将表明随着比率向上层适应性梯度移动变得更大和或扩散比率变小，这个生物的扩散被我们的模型预测接近于期望的理想自由栖息地的选择。其他的生物模型中生物被假设成沿着适应性梯度向上扩散已经在[3] P.R. Armsworth, J.E. Roughgarden, The impact of directed versus random movement on population dynamics and biodiversity patterns, Am. Nat. 165 (2005) 449–465.中研究了。【3，4】中两个种群模型被用来代替反应移流分布模型。在【19】中的分析方法和问题中，通过模型和分析解决问题是两种不同的方法----从以前那些文本中可以看出。

我们分析的部分动机是一种对理解在空间变化但时间不变的环境下演变的扩散的兴趣。在那种情况下它遵循McPeek和Holt【22】和区分非条件、有条件扩散之间的区别是有用的。无条件扩散是指扩散而不考虑环境或其他生物的存在。纯扩散和与物质的移流相关的扩散（例如由于转动和流动）都是无条件扩散。有条件扩散是指受环境或其他生物的存在影响的。它已经表明，在这个空间框架下明确的种群模型在空间变化看时间不变的环境下，只有无条件的扩散演化偏向缓慢的扩散。为什么无条件扩散室不利的是因为它导致了种群的分布和资源的分布的不匹配。但是，对于某些有条件的扩散类型，演化又是能够有利于更快的扩散如果他允许种群沿着资源更有效的方向扩散。这些结论考虑两个竞争对手的模型，它们采用不用的扩散策略，但其他生态相同，并且考察就入侵而言演化的稳定性。（一个策略被认为是进化上稳定的，如果这个种群用那种策略不能被一个使用不同策略的小种群入侵）。我们计划，在今后的工作中，从这个观点考虑理想自由分布。要做到这一点，我们需要理解一个单一种群使用理想自由扩散的行为；发展这个理解是本文的目标；进一步是的注意的是，导致了包含某些理想自由分布的特征的扩散过程已经被证明在离散扩散模型中是进化稳定的，见【10,25】。然而，同样应该被注意的是，在时间变化的模型中的系数或复杂的动态，更快的无条件扩散有时可能会更有利。其中一些现象和其他生态的部分，定向对抗随机移动和演化的扩散的影响，在两个种群的模型【3，4】中被研究。

1.2 研究问题

一个理想自由分布的关键想法是，个体们用这样一个方式—为了优化他们

的适应性，将它们定位。因此，在平衡水平上，在栖息地被占领的部分，所有生物将有相同的适应性并且这里讲没有个体的净运动，种群是恒定的。一个连续捕获这些特点的模型在【23】中被引进。假设一个种群有一个固有的人均增长率，m(x),该m在空间上不同但是经历增加的死亡率和或减少的繁殖成功率由于拥挤在整个环境一致得变化。如果种群密度被适当的缩放，个体的本地生殖适应性在x处，在同一密度u(x)的个体面前，是由

f(x,u)=m(x)-u(x)

给出的。

让F表示生物在占用栖息地Ω的部分的适应性。对一个固定总人口数U，种群的分布将由

u = _m(x)? F if m(x) > F ;

0 otherwise, 给出。其中F是尽可能符合条件的大。

?x??:u(x)?0??F??x??:u(x)?0)dx?U ?m(x?这些条件的第一只要总人口是守恒的。第二条件是通过，结合以前的密度公式u(x).他能够被用于定义F并且在u(x)>0的部分，通过观察它作为一个约束和最大F约束。在简单情况下，他可能找到F的显示的公式和u(x)>0的部分的U；见【23】。一个动态模型，支持平衡解，与这个在【14】中被引入的构想一致。这个模型有如下形式 on, 与无通量边界条件 on,

定义域是中的有界域，有光滑边界?Ω，n是在?Ω上的外向单位法向量，α是正常数，

用以衡量扩散强度的适应性梯度。

单一物种模型

在本文中，我们将考虑对上述模型的变化，包括人口的增长和沿着定向的适

应性梯度运动的扩散。自然要问，人口增长和扩散怎样相互作用。这是合理的假设：评定适应性梯度的过程，有瑕疵的梯度的跟踪，和对其他环境方面的反应可能造成一定量的随机运动。同时，通过将分布和人口动态包含进模型，我们能够研究过的模型相比较。在人口动态面前，纯理想自由扩散将预计导致一个平衡的人口分布，该分布中每个个体的适应性是0，因此，将没有进一步的人口增长。这相当于人口密度u(x)=m+(x),这里m+(x)表示m(x)的正向部分。如果m(x)被解释为描述资源的分布，那么有u=m+(x)意味着人口完全符合资源密度。我们将看到，随着生物向适应性梯度移动的趋势变大，这样的分布近似于对应的扩散模型的平衡。这将与包含运动梯度m(x)但不反应拥挤的模型的行为相对比。在这些模型中，生物的分布随着向梯度的移动速率变大，而趋向于变得集中在m(x)的局部最大值附近，见【12,13】

这个模型有如下形式

ut??????u??u?f(x,u)??uf(x,u)

(1.2)

in. (1.1)

无通量边界条件 on 这里

(1.3)

方程中u(x,t)表示单一种群的密度，而随机扩散系数是。用来衡量种群向适应性梯度移动的倾向，用f(x,u)表示。我们假设是正常数，是非负常数。Ω是一个上的有界域，边界是?Ω，n表示其上的外向单位法向量。在本文中我们假设m∈,，，且m在Ω上可取正。u(x,0)是连续的，非负的 ，不衡等零的。

两物种模型

最终我们计划研究，演化稳定的理想自由扩散与其他扩散策略的比较。要做到这一点，我们会考虑双物种模型，它们生态相同但是采用不用的扩散策略。这下，会导致一个系统形式

ut??????u??u?f(x,u?v)??uf(x,u?v) in. vt??????v??v?g(x,u?v)??vf(x,u?v)

in.

(1.4)

无通量边界条件

?u?f(x,u?v)?v?g(x,u?v)???u????v?0 on, (1.5)

?n?n?n?n

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Cantrell Chris Cosner Yuan Lou Chao Xie

This work extends previous work (Cantrell et al., 2008 [9]) on fitness-dependent dispersal for a single species to a two-species competition model. Both species dynamics, but one species adopts a combination of random and fitnessdependent dispersal and the other adopts random dispersal. Global existence of smooth solutions to the time-dependent quasilinear parabolic system is studied. When a single species a stable equilibrium that can approximate the spatial distribution predicted by the ideal free distribution (Cantrell et al., 2008 [9]). For the twospecies competition model, if one species analysis

shows that two competing species can coexist when one species intermediate tendency to move up its fitness gradient and the other species

This work extends our previous work [9] on fitness-dependent dispersal for a single species to a two-species competition model, with one species adopting a combination of random and fitnessdependent dispersal and the other adopting random dispersal. The model we considered in [9] u(x, t) represents the density of a single species with random diffusion coefficient μ, and α measures the tendency of the species to move upward along the gradient of the fitness of the species, measured by f (x, u). We assume that μ is a positive constant and α is a non-negative constant. Ω is a bounded region in RN with boundary ?Ω, and n denotes the outward unit normal vector on ?Ω. Throughout this paper we assume that m ∈ C2,γ (Ω) for some γ ∈ (0, 1) and m is positive somewhere in Ω, and u(x, 0) is continuous, non-negative and not identically zero in Ω. We briefly summarize some of the main results in [9] as follows:

? (Global existence in time) Suppose that μ > 0 and α _ 0. Then (1.1) u ∈ C2,1(Ω ×(0,∞)) ∩ C(Ω × [0,∞)).

? (Existence of positive steady state) If u = 0 is linearly unstable, then (1.1) Ω, then for large αμ, (1.1) of dispersal, a common approach, initiated by Hastings [24] for reaction– diffusion models, is to consider models of two populations that are ecologically identical but use different dispersal strategies. In general, using such a modeling approach would lead to a system of the form

?ut??????u??uf?x,u?v???uf?x,u?v????v??v?g(x,u,v)??vf(x,u?v) ?vt???????u??u?f?x,u?v???n????u??u?g?x,u,v???n?0? (1.3)

where f is as in (1.2), and g represents part of an alternate dispersal strategy. For example, g = 0

would correspond to unconditional dispersal of organisms by simple diffusion, g = m would correspond to advection up resource gradient without consideration of crowding, while g = ?(u + v) would correspond to avoidance of crowding without reference to resource distribution. We refer to reaction–diffusion models.

In this paper we will focus on system (1.3) with g = 0, i.e.,

?ut??????u??uf?x,u?v???uf?x,u?v?? ?vt???v?vf(x,u?v)????u??u?f?x,u?v???n??v?n?0? (1.4)

where the initial conditions u(x, 0) and v(x, 0) are non-negative and not identically zero in Ω, and μ, ν, α are all positive constants.

Our analysis is partially motivated by an interest in understanding the evolution of dispersal in spatially varying but temporally constant environments. In that context it is useful to follow McPeek and Holt [28] and distinguish between unconditional and conditional dispersal. Unconditional dispersal refers to dispersal without regard to the environment or the presence of other organisms. Pure diffusion and diffusion with physical advection (e.g. due to winds or currents) are examples of unconditional

dispersal. Conditional dispersal refers to dispersal that is influenced by the environment or the presence of other organisms. It shown that in the framework of spatially explicit population models on spatially varying but temporally constant environments with only unconditional dispersal that evolution favors slow leads to a mismatch between the distribution of population and the distribution

of resources. However, for certain types of conditional dispersal, evolution can

sometimes favor faster dispersal if that allows the population to track resources more two competitors that use different dispersal strategies but otherwise are ecologically identical, and examining the evolutionary stability of the strategies in terms

of invasibility. (A strategy is considered evolutionarily stable if a population using that strategy cannot be invaded by a small population using a different strategy.) We plan to consider ideal free dispersal from that viewpoint in future work. To do that, we need to understand well the behavior of a single species using ideal free dispersal; developing that understanding is the goal of this paper. Further it is worth noting that dispersal processes that result in patterns embodying certain features of the ideal free distribution shown to be

evolutionarily stable in discrete diffusion models; see [10,26]. However, it should also be noted that in models with temporal variation in the coefficients or complex dynamics, faster unconditional dispersal may the ecological effects of directed versus random movement and the evolution of dispersal are studied in the context of two-patch models in [3,4].

f(x,u)=m(x)-u(x)

Let F denote the fitness of organisms in the occupied part of the U the distribution of the population will be given by

u = _m(x)? F if m(x) > F ;

0 otherwise, where F is made as large as possible subject to the conditions

?x??:u(x)?0??F??x??:u(x)?0)dx?U ?m(x?The first of these conditions simply requires the total population to be conserved. The second condition is obtained by integrating the previous formula for the density u(x). It can be used to determine F and the region where u(x) > 0 by viewing it as a constraint and maximizing F subject to that constraint. In simple cases it is possible to find explicit formulas for F and the region where u(x) > 0 in terms of U; see [25]. A dynamic model which

supports equilibrium solutions corresponding to this formulation was introduced in [14]. That model , with the no-flux boundary condition

on,

where the in RN with smooth boundary ?Ω, n is the outward unit normal vector on ?Ω, and α is a positive constant that measures the strength of dispersal up the fitness gradient.

In the present paper we will consider a variation on the model of [14] that incorporates population growth and diffusion along with directed motion up the fitness gradient. It is natural to ask growth interacts with dispersal. It is reasonable to assume that the process of assessing the fitness gradient, imperfect tracking of that gradient, and responses to other aspects of the environment could lead to some amount of random movement. Also, by incorporating diffusion and population dynamics into the model, we can put it into a framework that allows us to compare it with other models that studied in the context of the evolution of dispersal would be expected to result in an

equilibrium distribution of the population in which the fitness of each individual will be zero, so that there will be no further population growth. That would correspond to a population density u(x) = m+(x),

where m+(x) denotes the positive part of m(x). If m(x) is interpreted as describing the distribution of resources, that the population perfectly matches the resource density. We will see that as the tendency of the organisms to move upward along fitness gradients becomes large such a distribution is approximated by the equilibria of the corresponding model with diffusion. This is in contrast with the behavior of models that incorporate movement up the gradient of m(x) but no response to crowding. In those models the distribution of organisms tends to become concentrated near local maxima of m(x) as the rate of movement up the gradient becomes large; see [12,13].

The model we will consider .

With on-flux boundary conditions

on

(1.2)

(1.1)

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