Global Stability and Oscillation of a Discrete Annual Plants Model
Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 156725, 18 pages
doi:10.1155/2010/156725
Research Article
Global Stability and Oscillation of
a Discrete Annual Plants Model
S. H. Saker1, 2
1
2
Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Correspondence should be addressed to S. H. Saker,
Received 26 September 2010; Accepted 2 November 2010
Academic Editor: Nicholas Alikakos
Copyright q 2010 S. H. Saker. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
The objective of this paper is to systematically study the stability and oscillation of the discrete
delay annual plants model. In particular, we establish some sufficient conditions for global stability
of the unique positive fixed point and establish an explicit sufficient condition for oscillation of the
positive solutions about the fixed point. Some illustrative examples and numerical simulations are
included to demonstrate the validity and applicability of the results.
1. Introduction
Most populations live in seasonal environments and, because of this, have annual rhythms
of reproduction and death. In addition, measurements are often made annually because
interest is centered on population changes from year to year rather than on the obvious
and predictable changes that occur seasonally. Continuous differential equations are not
well suited to these kinds of processes and data. Thus, practical ecologists have long
employed discrete-time difference equations for studying the dynamics of resource and pest
populations. In particular one can consider the difference equation
Nn 1 fNn,
n 0, 1, 2, . . . ,
1.1
as a measure of the population growth, where Nn 1 is the size of the population at time
n 1, Nn is the size of the population at time n, and the function fN is the densitydependent growth rate from generation to generation and in general it is a nonlinear function
of N. The skills in modelling a specific population’s growth lie in determining the appropriate
form of fN to reflect the known observations or the facts of the species under consideration.
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f(N)
3
2
1
0
0
1
2
3
4
5
N
Figure 1: Population model shape.
Density dependent is a dependence of per capita population growth rate on present and/or
past population densities. Hassell 1 proposed that the models of the population dynamics
in a limited environment are based on the following two fundamentals:
1 population have the potential to increase exponentially;
2 there is a density-dependent feedback that progressively reduces the actual rate of
increase.
In fact, in population dynamics, there is a tendency for that variable Nn to increase
from one generation to the next when it is small, and decrease when it is large. In 2
Cull showed that for population dynamics models the nonlinear function fN often has
the following properties: f0 0 and there is a unique positive fixed-point N such that
fN N, fN > N for 0 < N < N, and fN < N for N < N, and such that if fN has a
maximum NM in 0, N then fN decreases monotonically as N increases beyond N NM
such that fN > 0, see Figure 1.
In recent decades the dynamics of discrete models in different areas have been
extensively investigated by many authors. For contributions, we refer the reader to 2–17
and the references cited therein.
For population models of plants, Watkinson 18 assumed that the function fN
represents the number of seeds produced per parent plant which survived to flowering
in the next generation and reproduce seasonality and have effectively nonoverlapping
generations, even if a seed bank is present 18 . Using these assumptions Watkinson derived
some different forms of the function fN for seven different cases. For the annual plants
Watkinson 18 assumed that the density-dependent function is given by
fN :
λN
,
1 aNγ λmN
1.2
where λ is the growth rate and m represents the reciprocal of the asymptotic value of N
when the initial plant density tends to infinity and it is called the degree of self-thinning.
The parameter a has the dimension of the area and 1/a can be considered as the density
of plants at which mutual interference between individuals becomes appreciable and γ is
the density-dependent parameter where the biological significance is rather unclear. In 18
the author proposed that γ > 1, which reflects the fact that an increasing density leads to a
less-efficient use of the resources with a given area in terms of total dry matter population.
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3
Combining 1.1 and 1.2, we see that the Watkinson model of the annual plants is given by
the difference equation
Nn 1
λNn
,
γ
λmNn
aNn
1
n 0, 1, 2, . . . .
1.3
In this model the density-independent mortality is not included and the growth of the
population occurs only during the vegetative phase of the life cycle.
Watkinson in 18 assumed that density-independent mortality during the seed phase
of the life cycle can easily be incorporated by multiplying λ by the probability that a seed will
survive from the time of its formation to germination and establishment. Also in this model it
is clear that the past history of the population is ignored, that is, the growth of the population
is governed by a principle of causality, that is, the future state of population is independent
of the past and is determined solely by the present. In fact in a single species population
there is a time delay because of the time it takes a female animal or a plant to mature before
it can begin to reproduce. A more realistic model must include some of the past history of
population. Accordingly Kocić and Ladas 19 considered the model
Nn 1
λNn
,
1 aNn − 1γ mNn − 1
n 0, 1, 2, . . . ,
1.4
m ∈ 0, ∞,
1.5
and proved that if N−1 ≥ 0, N0 > 0 and
λ ∈ 1, ∞,
a ∈ 0, ∞,
γ ∈ 0, 1 ,
then limn → ∞ Nn N, where N is the unique fixed point of 1.4. Note that the assumption
γ ≤ 1 is different from the assumption γ > 1 that has been proposed by Watkinson 18 , which
reflects the fact that an increasing density leads to a less efficient use of the resources with a
given area in terms of total dry matter population.
In 11 the authors considered the general equation with two delays of the form
Nn 1
λNn
,
1 aNn − kγ λmNn − l
n 0, 1, 2, . . . ,
1.6
where
λ ∈ 1, ∞,
a, γ, m ∈ 0, ∞,
l, k ∈ {0, 1, 2, 3, . . .}.
1.7
The authors in 11, Theorem 6.3.1 proved that if
γ−1
Nγa 1 aN
mN l k /
1,
γ
1 aN λmN
1.8
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then every solution of 1.6 oscillates about N if and on (...truncated)
