A note on interacting populations that disperse to avoid crowding
A note on interacting populations that disperse to avoid crowding
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关于为避免拥挤而分散的相互作用人群的说明
DOI：
10.1090/qam/736508
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发表时间：
1984
影响因子：
0.8
通讯作者：
A. Pipkin
中科院分区：
文献类型：
--
作者：
M. Gurtin;A. Pipkin
在本文中，我们推导了为了避免拥挤而分散的种群的偏微分方程，特别注意个体之间分散的难易程度不一致的情况。我们建立了有限数量相互作用的生物群体和单个年龄结构群体的分散方程，并给出了后一个方程简化为前一个方程的条件。在所有情况下，这些方程都概括了经典的多孔流动方程——一个简并的抛物线方程，表现出无数有趣的效应。对于两个群体的特殊情况，我们推导出一个简单的解决方案，其中物种始终保持隔离。 1.基本方程。我们考虑 Rw 中 N 个生物类群的分散。例如，这些组可以对应于不同的生物物种或同一物种的不同年龄组。我们假设每个群体的扩散由三个函数描述：un(t,x)，人口密度，v„( t, x) 扩散速度，o„(f,x)，人口供应。场 un(t,\) 给出了群体 n 的个体数量，每单位“体积”，在时间 t\ 的位置 x，其在任何区域 R 上的积分给出了 R 中该群体的人口。从一点到另一点的人口流动由扩散描述速度 v„(f,x)，代表 n 组个体的平均速度。字段 <Jn(t, x) 给出了在 x 处提供个体的速率，例如，出生和死亡。这些字段被假定与平衡律2一致，du„/dt = -div(M„v„) + an。•1983 年 3 月 4 日收到。作者要感谢 M. Bertsch、D. Hilhorst 和 L. A.这项工作得到了国家科学基金会的支持。“关于扩散的一般讨论包含在 Levin [1976] 和 McMurtrie [1978] 的评论文章以及 Okubo [1980] 和 Nisbet 和 Gurney [1982] 的书中。2Cf„ 例如，[1977，Eq. (2.3)]。我们使用标准符号：lightface 字母标量、粗体字母是 R w 中的向量；v、div 和 A 分别表示 R1' 中的梯度、散度和拉普拉斯算子 ©1984 Brown University 88 MORTON E. GURTIN AND A. C. PIPKIN 我们将注意力限制在分散以避免拥挤的群体上。3 考虑到这一点，我们假设 y„ = -knvU，(1.1) 其中 kn 是一个称为分散性和 c/=2«» (1-2) n 总人口，以便每个群体（局部）向总人口的较低值分散，4 因此假设 °n = °n(u)> " = ("i, u2,...,uN)，基础偏微分方程的形式为 du„/dt = kndi\{unvU) + o„(u)。 (1.3) 由于我们关心所有 Rw 中的分散，我们只需邻接 (1.3) 形式为 K„(0,x) = W„(x) 的初始条件，且未指定。请注意，对于所有 n 和 2<t„(«)，kn — k 相同，仅 U 的函数 a(U)，(1.3) 产生 ^ = (t/2) + a(t/)，(1.4) 具有动力学的多孔流动方程。5 然而，当色散率 kn 时，会发生最有趣和不寻常的行为备注 1. 在关系式 (1.1), (1.2) 中，所有组都被分配相同的权重。事实上，我们可以用 U = 2Pnun, n 来代替 (1.2)，其中每个 /?n 都是严格正的常数。然后，定义 Wn = A,«n。我们再次恢复基本方程 (1.2) 和 (1.3)，但用 wn 替换。 3 Morisita [1950,1954]（水黾和蚁狮）、Ito [1952]（蚜虫）、Kono [1952]（稻象甲）的实地研究和实验证明了种群对传播的影响。Okubo [1980, §6] 和 Shigesada [1980] 对这些研究进行了讨论，另请参阅 Carl 的评论[1971] 关于北极地松鼠的扩散。4Cf。另请参阅 Busenberg 和 Travis [1983]，他们利用了 kn 独立于 n 的形式的假设。Gurney 和 Nisbet 给出了基于 (1.4) 的 5 个种群动态理论。 [1975, 1982] 和 Gurtin 和 MacCamy [1977]（另请参见 Shigesada [1980]）。有关多孔流动方程的一般研究，请参见例如 Oleinik [1965]、Aronson [1969]、Peletier [1981]。为了避免拥挤而分散的种群 89 备注 2 在某些情况下可能是合适的。允许色散系数 kn 为 un 和 U 的函数。在这种情况下，(1.3) 被替换为 0 = di\[unkn{un, £/)vt/] + on{u) 备注 3. 这里提出的类型的理论通常基于平衡定律。
In this note we derive partial differential equations for populations that disperse to avoid crowding, paying particular attention to situations in which the ease of dispersal is not uniform among individuals. We develop equations for the dispersal of a finite number of interacting biological groups and for a single age-structured group, and we give conditions under which the latter equations reduce to the former. In all cases the equations generalize the classical porous flow equation—a degenerate parabolic equation that exhibits a myriad of interesting effects. For the special case of two groups we deduce a simple solution in which the species remain segregated for all time. 1. Basic equations. We consider the dispersal1 of N biological groups in Rw. The groups may correspond, for example, to different biological species or to different age classes of the same species. We assume that the dispersal of each group is described by three functions: un(t,x), the population density, v„( t, x) the dispersal velocity, o„(f,x), the population supply. The field un(t,\) gives the number of individuals of group n, per unit "volume", at position x at time t\ its integral over any region R gives the population of that group in R. The flow of population from point to point is described by the dispersal velocity v„(f,x), which represents the average velocity of individuals of group n. The field <Jn(t, x) gives the rate at which individuals are supplied at x, for example, by births and deaths. These fields are assumed to be consistent with the balance law2, du„/dt = -div(M„v„) + an. •Received March 4, 1983. The authors would like to thank M. Bertsch, D. Hilhorst, and L. A. Peletier for valuable discussions. This work was supported by the National Science Foundation. 'General discussions of dispersal are contained in the review articles of Levin [1976] and McMurtrie [1978] and in the books of Okubo [1980] and Nisbet and Gurney [1982]. 2Cf„ e.g., [1977, Eq. (2.3)]. We use standard notation: lightface letters are scalars, boldface letters are vectors in R w; v, div, and A, respectivelu, denote the gradient, divergence, and laplacian in R1' ©1984 Brown University 88 MORTON E. GURTIN AND A. C. PIPKIN We limit our attention to groups which disperse to avoid crowding.3 With this in mind, we suppose that y„ = -knvU, (1.1) with kn a constant called the dispersivity and c/=2«» (1-2) n the total population, so that each group disperses (locally) toward lower values of total population,4 Thus assuming °n = °n(u)> " = ("i, u2,...,uN), the underlying partial differential equation takes the form du„/dt = kndi\{unvU) + o„(u). (1.3) Since we are concerned with dispersal in all of Rw, we need only adjoin to (1.3) initial conditions of the form K„(0,x) = W„(x), with un prescribed. Note that for kn — k the same for all n and 2<t„(«) a function a(U) of U alone, (1.3) yields ^ = (t/2) + a(t/), (1.4) the porous flow equation with kinetics.5 The most interesting and unusual behavior occurs, however, when the dispersivity kn varies from group to group, for then the groups disperse with different speeds. Remark 1. In the relations (1.1), (1.2) all groups are assigned equal weight. This assumption is not crucial. Indeed, in place of (1.2) we could take U = 2Pnun, n with each /?„ a strictly-positive constant. Then, defining Wn = A,«n. we once again recover the basic equations (1.2) and (1.3), but with w„ replaced by wn. 3 The effects of population on dispersal are demonstrated in the field studies and experiments of Morisita [1950,1954] (water striders and ant lions), Ito [1952] (aphids), Kono [1952] (rice weevils). These studies are discussed by Okubo [1980, §6] and Shigesada [1980], See also the remarks of Carl [1971] concerning the dispersal of arctic ground squirrels. 4Cf. the discussion of Gurtin and MacCamy [1977, §6]. See also Busenberg and Travis [1983], who utilize an assumption of the form (1.1) with kn independent of n. We saw this paper after completing our study. 5 Theories of population dynamics based on (1.4) have been given by Gurney and Nisbet [1975, 1982] and Gurtin and MacCamy [1977] (see also Shigesada [1980]). For general studies concerning the porous flow equation see, for example, Oleinik [1965], Aronson [1969], Peletier [1981]. POPULATIONS THAT DISPERSE TO AVOID CROWDING 89 Remark 2. In some instances it might be appropriate to allow the dispersivities kn to be functions of un and U. In this instance (1.3) is replaced by 0 = di\[unkn{un, £/)vt/] + on{u). Remark 3. Theories of the type presented here are often based on the balance law