舆论摘要：两类带各别反馈项的捕食-食饵模子解的本质领会
生态学中的捕食-食饵模子的接洽在往日的几十年里有了很好的截止，而且对具备分散项的捕食-食饵模子的接洽获得了很多新的截止.暂时，人们应用反馈分散方程表面来接洽生态范围中的数学模子,已变成一个十分抢手的课题.二维捕食-食饵体例中典范的模子是Volterra-Lotka模子,如次：$$left{ egin{array}{l} frac{partial u}{partial t}-D_{1}Delta u=u(a_{1} - b_{1}u - c_{1}v) , xin Omega, t>0, frac{partial v}{partial t}-D_{2}Delta v=v(a_{2} + b_{2}u - c_{2}v), xin Omega, t>0, Bu = Bv = 0 , x in partialOmega, t>0, u(x,0)=u_0(x)geq 0,otequiv 0; v(x,0)=v_0(x)geq 0,otequiv 0, xin Omega . end{array} ight.eqno{(0)}$$ 个中$Omega subseteq R^{n}(ngeq0)$是边境$partialOmega$充溢润滑的有界地区$;Delta$为Laplace算子; $u$, $v$辨别表白地区$Omega$中食饵和捕食者的密度,参数$a_{i}$, $b_{i}$, $c_{i}$, $D_{i}$ (i=1,2) 均为平常数,而边值参数$B$的情势为 $$Bu=left{ egin{array}{l} u, $或$ frac{partial u}{partial u}+b_{0}u, end{array} ight.$$ 这边的$frac{partial u}{partial u}$表白沿单元外法线的方引导数,$b_{0}(x)geq 0(x in partialOmega)$.人们对上头体例更多的关心是两物种是否并存大概是一物种连接存在而另一物种消失.从数学观点来领会,即当$tightarrow+infty$时,上头方程的解 $(u,v)$恒为正数仍旧$uightarrow0$或$vightarrow0$.而在计划的进程中，很简单看出，并存题目与平稳态体例的正解生存性精细关系,解的循序渐进动作与平稳态解的本质如宁静性联系出色.所以越发凸现出平稳态题目接洽的要害性,比方说解的生存性、宁静性等题目均为人们关心的热门.正文在普遍Volterra-Lotka模子的普通上,计划了带Holling Type Ⅲ 反馈分散项和Beddington-DeAngelis 反馈项的捕食体例,辨别参观了如次三个简直的捕食模子：$$left{ egin{array}{l} frac{partial u}{partial t}-d_{1}Delta u=au- bu^{2} - frac{alpha u^{2}v}{eta^{2}+u^{2}}, xin Omega, t>0, frac{partial v}{partial t}-d_{2}Delta v=-c v+ frac{kalpha u^{2}v}{eta^{2}+u^{2}}, xin Omega, t>0, frac{partial u}{partialu} = frac{partial v}{partialu} = 0, x in partialOmega, t>0, u(x,0)=u_0(x)geq 0; v(x,0)=v_0(x)geq 0 , xin Omega . end{array} ight.eqno{(0.1)}$$ $$left{ egin{array}{l} frac{partial u}{partial t}-Delta u=u(a- u - frac{buv}{eta^{2}+u^{2}}), xin Omega, t>0, frac{partial v}{partial t}-Delta v=v(c-v+frac{du^{2}}{eta^{2}+u^{2}}), xin Omega, t>0, k_{1}frac{partial u}{partialu}+u=0, k_{2}frac{partial v}{partialu}+v = 0, x in partialOmega, t>0, u(x,0)=u_0(x)geq 0,otequiv 0; v(x,0)=v_0(x)geq 0,otequiv 0 , xin Omega . end{array} ight.eqno{(0.2)}$$ $$left{ egin{array}{l}frac{partial u}{partial t}-Delta u=ru-ku^{2}- frac{alpha uv}{a+bu+cv}, xin Omega, t>0, frac{partial v}{partial t}-Delta v=dv+frac{eta uv}{a+bu+cv}, x in Omega, t>0, u=v=0, xin partialOmega, t>0, u(x,0)=u_0(x)geq 0,otequiv 0; v(x,0)=v_0(x)geq 0,otequiv 0, x in Omega. end{array} ight.eqno{(0.3)}$$ 个中$Omega subseteq R^{n}(ngeq0)$是边境$partialOmega$充溢润滑的有界地区;系数参数均为平常数, $frac{alpha u^{2}v}{eta^{2}+u^{2}}$为Holling Type Ⅲ反馈分散项, $frac{uv}{a+bu+cv}$是Beddington-DeAngelis反馈分散项. 正文分三局部对这三个简直捕食模子解的本质举行了计划. 第一章计划了体例(0.1)在分散系数不沟通的情景下非负常数解的宁静性,应用极值道理,比拟道理及Harnack不等式决定了体例(0.1)正平稳态解的少许先验估量,而后计划了特殊数解的不生存性,特殊数正平稳态解的全部生存性及特殊数正平稳态的分别解.第二章计划了体例(0.2)平稳态体例解的本质,运用锥不动点目标和同伦静止性计划了体例特殊数并存解的生存性与不生存性, 而且参观了平稳态体例的限制分别解的生存性、独一性及解的宁静性.第三章计划了一类带Beddington-DeAngelis反馈项的捕食体例在Drichlet边境前提下平稳态解的本质.运用比拟道理作领会的少许先验的估量,应用限制分别表面、特性值扰动表面和全部分别表面.重要获得了方程$(0.3)$的限制分别解生存的充溢前提及解的宁静性, 并给出了全部分别解生存性及其解的简直走向.
