Abstract:
How intrinsic factors (such as intrinsic growth rate) and external factors (such as migration) affect the long-range dynamic behavior of population density of a single population has been well studied (1-7). However, when a more practical model of the behavior of linked populations is considered, strange phenomena, such as jumping, complementation, period reversal, sensitivity to initial population density, will appear. Such phenomena are more likely to occur in nature.
Introduction:
Since Robert May pointed out in 1976 that simple ecological models can yield very complicated results(1), the study on the dynamic behavior of population density became more and more thorough. In the beginning, May, Hassel, etc, built some models of an isolated single population (i.e. only intrinsic factors are considered), pointing out that such models can give complicated results, such as period cycling and chaos.(1,5) Then Sinha, Stone, etc, put some external factors such as migration into consideration, pointing out that the behavior of a single population can be easily changed by these factors. For example, immigration may easily suppress chaos.
Sinha and Stone, etc, used a constant to represent migration(2,4,7). This, however, is not likely to happen in nature. So I took the source of the migration into consideration, i.e. the two linked populations are both considered. In such cases, migration occurs between the two populations.
In this paper, first, I will build and analyze a model to describe the dynamic behavior of the two linked population densities; then investigate the effects of the intrinsic growth rate on the behavior and compare such effect with that in a single population model; thirdly, the sensitivity of the behavior to the initial density of the two populations will be considered and at last I will analyze the sensitivity to intrinsic growth rate and the emigration critical point, the density of a population above which emigration occurs.
Text:
1.
process of population growth
Two linked, synchronized populations are considered. For each of them, we can divide its growth into two phases. 1) intrinsic growth without migration; 2)migration between the populations. See figure 1. The density after the phase 1 and before phase 2 is considered(11).
2.
model and analysis
To simplify the problem, we make the following assumptions:
1)No other population is linked to the linked ones. The emigration of one population is the immigration of the other.
2)The environments of the two populations are identical except the geological sites. For example, the values of their maximum capacities are equal.
3)The two populations share the same intrinsic growth rate and the emigration critical point.
The following difference equations are employed to describe the dynamic behavior of two population density based on the above assumptions:
Xn+1 = f[Xn – m(Xn) +
m(Yn)]
Yn+1 = f[Yn – m(Yn) + m(Xn)] (1)
where
f(X) = r·X·exp[-r·X/(Nmax·e)] (2)
m(X) = sqrt( X 2– m02) X>m0
0 X<=m0 (3)
The relationship between the density of the nth and that of the (n+1)th is described in equation (1), where emigration---m(X) is determined by equation (3). The parameter r represents intrinsic growth rate, Nmax represents the carrying capacity of the environment. Without loss of the generality we give Nmax the value of 10. m0 represents the emigration critical point, when the density of a population is above this value, emigration will occur. e is the base of natural logarithm.
The reason for assuming such a
format to f(X) is that the format has
its own character. The function has the
same single maximum Nmax no matter
how much the intrinsic growth rate r is.
This is more practical than equations such as the logistic one(5):
f(X) = rX(1-X) (4)
because in nature, the density of a population will always reach the environment’s carrying capacity, as long as the previous generation’s density is suitable and no matter how much the intrinsic growth rate r is.
Equation (3) is also more practical than just a constant, which is employed as the migration rate in many models(2,4,7). Emigration only occurs when the density of the population reaches a certain value, which we call the emigration critical point. After this point, emigration will increase with the rise in the density of the population.
3.
results
3.1 the effects of intrinsic growth rate on the
long range dynamic behavior of the population’s density
To explore such effects the long range limits of the two populations’ density to a series of r is calculated by computer. The relationship between the limits and r is given in the figure 2. The model is described the following difference equation:
Xn+1 = f(Xn) (5)
From the figure we can see that the dynamic behavior of the two linked populations’ density is much more complicated than that of the single one. First, figure 2(b) shows some characters such as bifurcation seen from figure 2(a), but in fact the two populations occupy on branch each. I call this feature complementation. Why such characters occur is not know. Secondly, the way to chaos by period doubling easily seen in single population model (figure 2a) is not likely seen in two linked population models. Thirdly, figure 2b shows jumping, which means the behaviour changes immensely at the jumping points. Fourthly, long range low period cycling appears between chaos, where in figure 2a such phenomena only occur before chaos. At last, period-doubling reversals seen in figure 2c do not occur in figure 2a.
3.2 sensitivity to initial density of the two
population (X0,Y0)
Now the sensitivity to initial densities of the two populations is considered. Given a certain growth rate r(=11.6) and emigration critical point m0(=5.0) The period number to all the possible initial density(X0,Y0) is calculated. Using different colous to represent different period numbers, an figure in X0-Y0 plane is got. See figure 3.
We can see that it is a fractal picture. In some ranges, such as one near the diagonal, the period number will change easily with a very small change in initial densities. So we can draw the conclusion that the behavior is sensitive to the initial densities of the two populations. This is not seen in single population models, the long range behavior is not sensitive to the initial density of the single population.
3.3 sensitivity to emigration critical point m0 and intrinsic growth
rate r
By fixing a certain initial density of the populations (X0=2.0,Y0=3.0) and varying the emigration critical point m0 and intrinsic rate r, we can get a picture in m0-r plane with the same method described in 3.2. The relationship between the dynamic behavior and (r,m0) is clearly seen in this projection. See Figure 4. As we can see, the behavior is very sensitive to r and m0. Fixing a certain intrinsic growth rate r(say,20), the very small change of m0 will lead to a vast change of the behavior. The same is to r by fixing m0. This can also be seen in single population models which show the sensitivity to intrinsic growth rate r and migration rate.
Conclusion and Discussion:
Because a majority of populations are linked in nature, the conclusions drawn based on the model of single population are not likely to occur in the field. Many results given by the model of linked populations are different from ones by single population models. For example: jumping, complementation，sensitivity to initial density, etc. These phenomena are more likely to occur in nature.
But this is far from sufficient because in nature, generally more than two populations are linked and they form a network. What is the behavior of such a network is necessary to be studied.
Also, only long range behavior is considered in most studies on theoretical population density dynamics. As we see in nature, transient behaviors are also important so research on it demands attention.
Reference:
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