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, [1]), 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. [2]. 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 [1]
and the third is a Ricker nonlinearity, [3]. 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.
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 [5]. 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. [8], 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.
