تجزیه و تحلیل سیستم های کنترل بازخورد تراکم دینامیک جمعیت وابسته
|کد مقاله||سال انتشار||مقاله انگلیسی||ترجمه فارسی||تعداد کلمات|
|28097||2012||7 صفحه PDF||سفارش دهید||محاسبه نشده|
Publisher : Elsevier - Science Direct (الزویر - ساینس دایرکت)
Journal : Systems & Control Letters, Volume 61, Issue 2, February 2012, Pages 309–315
We use feedback control methods to prove a trichotomy of stability for nonlinear (density dependent) discrete-time population dynamics defined on a natural state space of non-negative vectors. Specifically, using comparison results and small gain techniques we obtain a computable formula for parameter ranges when one of the following must hold: there is a positive, globally asymptotically stable equilibrium; zero is globally asymptotically stable or all solutions with non-zero initial conditions diverge. We apply our results to a model for Chinook Salmon.
The familiar feedback control design for single-input, singleoutput discrete-time systems xt+1 = Axt + but , yt = cT xt , with nonlinear output feedback u = f ( y), leads to a closed-loop system xt+1 = Axt + bf (cT xt ). (1.1) Feedback descriptions of this type arise also in nonlinear population dynamics. For example, the population dynamics of a fish species (e.g., p. 316–323, ), with density dependent survival of eggs, can be modelled in this form. In this application, the state xt describes the population structure of the fish at time t, with population structure determined by discrete, developmentalbased stage classes. The right hand side of (1.1) captures two fundamental biological processes—survival/growth and fecundity of fish in each size class. In the case of (1.1), A models linear demographic transition rates, whilst the term bf (cT xt ) picks up specific nonlinear, density limited transitions. The matrix A is nonnegative (all entries of A are non-negative), cT xt is a nonnegative weighted population density and the non-negative vector b describes the population structure of new-born fish. Density dependence is captured by f , which determines the nonlinear relationship between egg production and survival to one-year old fish. Similar nonlinear (i.e. density dependent) models arise when considering the population dynamics of monocarpic plants, for example Platte Thistle, see Rose et al. . In this case, the nonlinearity captures the density dependence of seedling establishment. Typical density dependences which are used in population dynamic models are: f (y) = βyα with α ∈ (0, 1) and β > 0; f (y) = Vy K + y with V > 0 and K > 0; f (y) = y exp(−βy), β > 0. The first is a power-law type nonlinearity, the second is of the so-called Beverton–Holt (equivalently Michaelis–Menten) type  and the third is a Ricker nonlinearity, . In the first two cases the nonlinearity f is monotone, but the third is not and f ( y) has a maximum. Hence the nonlinear model (1.1) is a candidate for density dependent population dynamics of both flora and fauna. Whilst the feedback structure (1.1), is quite familiar in systems theory, this feedback structure has not been widely exploited in population biology.The paper is organised as follows: In Section 2 we formulate the assumptions about system (1.1) and state our main result, namely Theorem 2.1. Section 3 is devoted to a proof of this main result via a sequence of lemmas. This section also contains an extension of this main result to the case when the underlying system in not monotone. In Section 4 we illustrate our main results with two examples.
نتیجه گیری انگلیسی
We have used feedback control methods, specifically comparison results and small gain techniques, to characterise a trichotomy of stability for nonlinear (density dependent) population dynamics. We have focused on populations modelled in discrete size or stage classes where the natural state space is the cone of nonnegative vectors in Rn. Our results do not require the system to be monotone and so our results generalise trichotomy of stability results in . The characterisation of the trichotomy requires knowledge of a steady state gain G(1) and sector-type constraints on f which will be checkable without precise knowledge of the system. Determining stability type from poor data is important in ecological applications because paucity of, and uncertainty in, data is the norm. We apply our results to a model of Chinook Salmon. In this case, ranges of parameters where the various limiting behaviours occur can be characterised by the population’s reproductive rate. In Rebarber et al. , we give versions of our results for Integral Projection Models (see [9,10]) which are relevant for populations, such as plants, that are best described by continuous size structures.