【正文】 ˇ cik, The evolution of dispersal rates in a heterogeneous timeperiodic environment, J. Math. Biol. 43 (20xx) 501– 533. 具適應(yīng)性的人口疏散模型的 整體解 25 致謝 經(jīng)過幾個(gè)月的忙碌和工作，本次畢業(yè)論文已經(jīng)臨近尾聲。 我們先給出問題 ()的一個(gè)局部解存在性結(jié)論 。kk ku u u? ?? ? ? ????. 現(xiàn)在記區(qū)間 (0, )IT? , TQ 表示 1nR? 中的柱體 I?? . 我們引入空間 ,( ) 2()kk TCQ???? . 對(duì)于 (,)uxt , 我們引入如下半模 : 具適應(yīng)性的人口疏散模型的 整體解 11 ? ? ? ?。這種方法已經(jīng)被用在【 11,12,16,25】中。自然要問，人口增長(zhǎng)和擴(kuò)散怎樣相互作用。因此，在平衡水平上，在棲息地被占領(lǐng)的部分，所有生物將有相同的適應(yīng)性并且 這里講沒有個(gè)體的凈運(yùn)動(dòng)，種群是恒定的。在【 19】中的分 析方法和問題中，通過模型和分析解決問題是兩種不同的方法 從以前那些文本中可以看出。 具適應(yīng)性的人口疏散模型的 整體解 1 具適應(yīng)性的人口疏散模型的整體解 摘要 在人口疏散中，采取怎樣的方式疏散 (擴(kuò)散 )人口更為有效是一個(gè)非常重要的問題。 我們分析的部分動(dòng)機(jī)是一種對(duì)理解在空間變化但時(shí)間不變的環(huán)境下演變的擴(kuò)散的興趣。一個(gè)連續(xù)捕獲這些特點(diǎn)的模型在【 23】中被引進(jìn)。這是合理的假設(shè)：評(píng)定適應(yīng)性梯度的過程，有瑕疵的梯度的跟蹤，和對(duì)其他環(huán)境方面的反應(yīng)可能造成一定量的隨機(jī)運(yùn)動(dòng)。使用這種建模方法，在理想自由擴(kuò)散背景下，會(huì)導(dǎo)致一個(gè)系統(tǒng)形式 ? ? ),(),( vuxufvuxfuuu t ????????? ?? in ? ???? ,0 . ? ? ),(),( vuxvfvuxgvvv t ????????? ?? in ? ???? ,0 . () 無通量邊界條件 0),(),( ?? ??????? ????? n vuxgvnvn vuxfunu ???? on ? ????? ,0 , () 這里 f同 ()， g 代表替換的擴(kuò)散策略的一部分。s u p ( , )Txk Q ktu u t??? ? ???, ? ? ? ?。 定理 假設(shè) m(x), ? ? ? ?1,0,)(),( 200 ??? ? ??Cxvxu 則存在某個(gè) ? ???? ,0maxT ,使得問題()存在唯一的解 (u,v)滿足 ? ? ? ?? ? ? ? ? ?? ?m a x22,2 ,0, TCtxvtxu ??? ?? ?? 進(jìn)一步，如果 ??maxT 則有 ? ? ? ? ??????Ltu ,當(dāng) maxTt? 證明： 該局部存在性結(jié)果是經(jīng)典結(jié)論，證明參見參考文獻(xiàn) [1][2] 具適應(yīng)性的人口疏散模型的 整體解 14 我們?cè)噲D證明 ()的古典解在整個(gè)時(shí)間 ? ??,0 上存在 ，根據(jù)定理 (),只要證明 ()存在某個(gè)常數(shù) C0，使得對(duì)任何 T0,都有 ? ? ? ? CtuL ?? ??,對(duì) ? ?Tt ,0?? () 而 ()的證明是建立在一些引理的基礎(chǔ)上的。作為一個(gè)本科生，由于經(jīng)驗(yàn)的匱乏，難免有許多考慮不周全的地方，如果沒有導(dǎo)師的督促指導(dǎo)，同學(xué)的支持鼓勵(lì)，想要獨(dú)自完成這個(gè)論文是相當(dāng)困難的 . 在這里，我要感謝我的導(dǎo)師陶有山老師 .老師平日里工作繁 忙，但在我做畢業(yè)論文的每個(gè)階段，從論文開題到查閱資料，中期檢查，后期撰寫與修改等整個(gè)過程中都給予了我悉心的指導(dǎo) .一絲不茍的工作作風(fēng)，求真務(wù)實(shí)的態(tài)度，踏實(shí)的鉆研精神，不僅授我以文，而且教我做人，雖僅歷時(shí)數(shù)月，卻給我受益無窮 . 在此表達(dá)對(duì)陶老師的衷心感謝！ 最后感謝東華大學(xué)四年來對(duì)我的栽培，同時(shí)我也快樂地度過了本科四年的學(xué)習(xí)生活 . 具適應(yīng)性的人口疏散模型的 整體解 26 原文及譯文 Random dispersal versus fitnessdependent dispersal Robert Stephen Cantrell Chris Cosner Yuan Lou Chao Xie This work extends previous work (Cantrell et al., 20xx [9]) on fitnessdependent dispersal for a single species to a twospecies petition model. Both species have the same population dynamics, but one species adopts a bination of random and fitnessdependent dispersal and the other adopts random dispersal. Global existence of smooth solutions to the timedependent quasilinear parabolic system is studied. When a single species has a strong tendency to move up its fitness gradient, it results in a stable equilibrium that can approximate the spatial distribution predicted by the ideal free distribution (Cantrell et al., 20xx [9]). For the twospecies petition model, if one species has strong tendency to move up its fitness gradient, such approximately ideal free dispersal is advantageous relative to random dispersal. Bifurcation analysis shows that two peting species can coexist when one species has only an intermediate tendency to move up its fitness gradient and the other species has a smaller random dispersal rate. 1. Introduction This work extends our previous work [9] on fitnessdependent dispersal for a single species to a twospecies petition model, with one species adopting a bination of random and fitnessdependent dispersal and the other adopting random dispersal. 具適應(yīng)性的人口疏散模型的 整體解 27 The model we considered in [9] has the form ? ?? ? ? ?? ?? ???? ???????????? 0, ,nuxfuu uxufuxfuuu t ?? ?? () Where ? ???? ,0R , and ? ? ? ? uxmuxf ??, () The function 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 nonnegative 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, nonnegative 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 () has a unique solution u ∈ C2,1(Ω (0,∞)) ∩ C(Ω [0,∞)). ? (Existence of positive steady state) If u = 0 is linearly unstable, then () has at least one positive steady state. Note that if _Ω m 0, u = 0 is linearly unstable for any μ 0 and α _ 0. ? (Global attractor) If m 0 in Ω, then for large α/μ, () has a unique positive steady state which is also globally asymptotically stable. To study the evolution of dispersal, a mon 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 ? ?? ? ? ?? ?? ?? ? ? ?? ????????????????????????????????0,),(),(,nvuxguunvuxfuuvuxvfvuxgvvvvuxufvuxufuutt???????? () where f is as in (), and g represents part of an alternate dispersal strategy. For example, g = 0 具適應(yīng)性的人口疏散模型的 整體解 28 would correspond to unconditional dispersal of anisms 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 [3–5,7–11,13,14,17,23,27,30–32,42,38,46] for recent progress in this direction for reaction–diffusion m