Colombo E H, Anteneodo C
Departament of Physics, PUC-Rio, Rio de Janeiro, Brazil.
Departament of Physics, PUC-Rio, Rio de Janeiro, Brazil; Institute of Science and Technology for Complex Systems, Rio de Janeiro, Brazil.
J Theor Biol. 2018 Jun 7;446:11-18. doi: 10.1016/j.jtbi.2018.02.030. Epub 2018 Feb 27.
A key issue in ecology is whether a population will survive long term or go extinct. This is the question we address in this paper for a population in a bounded habitat. We will restrict our study to the case of a single species in a one-dimensional habitat of length L. The evolution of the population density distribution ρ(x, t), where x is the position and t the time, is governed by elementary processes such as growth and dispersal, which, in standard models, are typically described by a constant per capita growth rate and normal diffusion, respectively. However, feedbacks in the regulatory mechanisms and external factors can produce density-dependent rates. Therefore, we consider a generalization of the standard evolution equation, which, after dimensional scaling and assuming large carrying capacity, becomes ∂ρ=∂(ρ∂ρ)+ρ, where μ,ν∈R. This equation is complemented by absorbing boundaries, mimicking adverse conditions outside the habitat. For this nonlinear problem, we obtain, analytically, exact expressions of the critical habitat size L for population survival, as a function of the exponents and initial conditions. We find that depending on the values of the exponents (ν, μ), population survival can occur for either L ≥ L, L ≤ L or for any L. This generalizes the usual statement that L represents the minimum habitat size. In addition, nonlinearities introduce dependence on the initial conditions, affecting L.
生态学中的一个关键问题是种群能否长期生存还是走向灭绝。这就是我们在本文中针对有限栖息地中的种群所探讨的问题。我们将研究限制在长度为L的一维栖息地中单一物种的情况。种群密度分布ρ(x, t)（其中x是位置，t是时间）的演化由诸如生长和扩散等基本过程所支配，在标准模型中，这些过程通常分别由恒定的人均增长率和正态扩散来描述。然而，调节机制中的反馈和外部因素会产生密度依赖率。因此，我们考虑标准演化方程的一种推广形式，在进行量纲缩放并假设承载能力很大之后，该方程变为∂ρ = ∂(ρ∂ρ) + ρ，其中μ, ν∈R。这个方程由吸收边界补充，模拟栖息地外部的不利条件。对于这个非线性问题，我们通过解析得到了种群生存的临界栖息地大小L的精确表达式，它是指数和初始条件的函数。我们发现，根据指数(ν, μ)的值，种群生存可能在L ≥ L、L ≤ L的情况下发生，或者对于任何L都可能发生。这推广了通常认为L代表最小栖息地大小的说法。此外，非线性引入了对初始条件的依赖性，从而影响L。
