On positive solutions of some pairs of differential equations
On positive solutions of some pairs of differential equations
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DOI：
10.1090/s0002-9947-1984-0743741-4
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发表时间：
1984-02
影响因子：
1.3
通讯作者：
E. N. Dancer
中科院分区：
文献类型：
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作者：
E. N. Dancer
本文是文[6]的继续，文[6]研究了由Conway，Gardner和Smoller [4]引入的一对微分方程的正解。他们的方程是生物学中人口问题的方程。其重要性将在第141页进一步讨论。本文的目的是得到Conwat-Gardner-Smoller方程的进一步结果，并说明如何修改[6]中的方法，以精确地判定经典的捕食-食饵系统（如Blat和Brown [2]）何时有严格正解.我们还展示了如何可以应用到其他一些问题的方法。特别地，我们的结果回答了Conway [S]中的一些问题。其次，我们证明了[6]中发展的渐近方法如何用于研究竞争种群问题严格正解的存在性和唯一性。我们的研究结果表明，这是一个更复杂的问题比捕食者-食饵问题。(We得到了一个条件，除了少数特殊情况外，该条件是存在严格正解的充分必要条件。然而，这个条件是复杂的，而且相当含蓄。）第三，证明了当d很小且n = 1时（即常微分方程的情形）Conway-Smoller系统严格正解的唯一性。这部分回答了[4]中的一个问题。我们的证明是一个局部的结果，可以用于其他情况。例如，它可以用来表明，在某些“反社会”非线性情况下（我们的符号遵循[4，第4节]），对于小d，恰好有两个严格正解。我们还展示了我们的方法如何适用于更一般的捕食-被捕食问题。在第一节中，我们找到了捕食-食饵系统严格正解存在的充分必要条件;在第二节中，我们研究了竞争种群模型，通过迭代和渐近方法。最后，在第3节中，我们证明了我们的唯一性结果。在附录中，我们简要讨论了如何将第3节的方法应用于更一般的模型。
This paper is a sequel to the author’s paper [6], which studied positive solutions of a pair of differential equations introduced by Conway, Gardner, and Smoller [4]. Their equations were equations for population problems in biology. Their significance is discussed further in 141. The purpose of the present paper is to obtain further results for the Conwat-Gardner-Smoller equations and to show how the methods in [6] can be modified to decide exactly when the classical predator-prey system (as in Blat and Brown [2]) has a strictly positive solution. We also show how the methods can be applied to a number of other problems. In particular, our results answer some questions in Conway [S]. Secondly, we show how the asymptotic methods developed in [6] can be used to study the existence and uniqueness of strictly positive solutions of competing species problems. Our results suggest that this is a much more complicated problem than the predator-prey problem. (We do obtain a condition which is, except for a few special cases, necessary and sufficient for the existence of a strictly positive solution. However, the condition is complicated and rather implicit.) Thirdly, we prove the uniqueness of the strictly positive solution in the Conway-Smoller system if d is small and n = 1 (that is, the case of ordinary differential equations). This partially answers a question in [4]. Our proof is a local result and can be used in other situations. For example, it can be used to show that, in certain cases of “asocial” nonlinearities (where our notation follows [4, Sect. 4]), there are exactly two strictly positive solutions for small d. We also show how our method applies to more general predator-prey problems. In Section 1, we find necessary and sufficient conditions for the existence of strictly positive solutions of predator-prey systems; in Section 2, we study competing species models, by iteration and asymptotic methods. Finally, in Section 3, we prove our uniqueness results. In an appendix, we briefly discuss how the methods of Section 3 apply to more general models.