The emigration critical value and the initial population size in a two-linked-population model
Consider a single population obeying the following difference equation if there is no migration:
Nt+1 = f(Nt) = rNt(1-Nt) (1)
where Nt+1 and Nt are the population sizes at generation t+1 and t respectively. r is a parameter, representing intrinsic growth rate.
Emigration occurs when the population size reach a certain value, which we call
the emigration critical value, represented by m0. And the emigration size is given by:
m=m(x)=N-m0 (2)
where m is the emigration size and N is the population size of current generation. Attention: if N is less than
m0, there is no emigration.
Now consider two populations linked by migration. We assume each of them has two phases: (1) growth without emigration and (2) migration without growth
(Fig.1). We also assume there is no other population linked to the above them. So the two populations obey:
xt+1=f[xt-m(xt)+m(yt)]
yt+1=f[yt-m(yt)+m(xt)] (3)
where xt+1 and yt+1 represent the two populations' sizes at generation t+1, respectively; and
xt, yt, the sizes at generation t.
There are two parameters, m0 and r, in this model and they will affect the population dynamics. There are also another two parameters,
x0 and y0, that probably matter.
How the critical value, m0, affects the long term dynamics of the two populations?
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(1)The population long term dynamics, an example
First let's see a (typical) dynamics of this model. We set the parameters as the following:
x0=0.2
y0=0.3
m0=0.6
We will see how the stable equilibrium change as the intrinsic growth rate increases, as shown in
Fig.2. The two populations have the same 1-periodic stable point when r is small. Then, as r increases (3.00<r<3.44), they fluctuate in the same way (2-periodic cycling). They continue fluctuating but in different ways where r>3.44 (2,4,8,...-periodic cycling and finally to chaos).
So a typical dynamics is such: (in a figure where both two population stable points are plotted) as intrinsic growth rate increases, the stable states will change and the number of them will double (bifurcation) and finally to chaos. Such dynamics is just like that of an isolated single population. But if only one population is displayed, we will see a somewhat fragmentary diagram. The phenomenon that integrity occurs only when both populations' stable points are plotted is called
complementation.
(2)As m0 changes, the dynamics changes
Then what if m0 changes?
From the series of figures in Fig.3 we clearly see that m0 does affect the dynamics, especially in the complementation region.
m0=0.5 seems a somewhat symmetric value. From 0.1 to 1.2 there is a trend of regular→unregular→regular→unregular→regular. The figures where
m0=0.4 and 0.9 show apparent period doubling reversals, though not smoothly. This means under such
m0, further increase r will decrease the fluctuate extent, or in other words, migration makes the two populations more stable, compared migration-free cases.
(3)m0 changes the periods greatly
Another approach is to determine the period of stable points as m0 changes. Because r also matters, we plot in
r-m0 plane. See Fig.4. It is clearly seen from
Fig.4a that when r ranges from 4 to 6,
m0 changes the period greatly. There are even more details in this range, as we amplify this range in
Fig.4b and Fig.4c. I only consider the case where the period is less than 10 and all those above 10 is plotted white, and thus lose much information. But the figure still shows great complexity, and some fractal-like characters (there is always details under every scale). This indicates that in some ranges, even if you just change
m0 a very little bit, the period, and thus the dynamics, will change greatly. In such ranges, the dynamics is sensitive to
m0. The area where 4<r<5 and 0.6<m0<1.0 is such a range. It seems that
m0 makes no difference where r<3.
Also, from this figure we see m0=0.5 is really a somewhat symmetric value. The yellow semicircular and some complex patterns aside are somewhat symmetric.
A note is needed to the white lines (r=1) and (r=3). In fact the two lines should be yellow (period 1). This is because we can only simulate a limited, though large, number of steps by computer, and assume the number is large enough to reach the stable point. But when r is near 1 or 3, the speed to the stable point is very slow and thus the computer decides it an unperiodic process.
(4)How m0 changes the stable point
A direct way is treating m0 as a parameter and draw a figure just using the same method of drawing the bifurcation diagram. See
Fig.5. In Fig.5b, the dynamics is really complicated. There is period doubling, period doubling reversals, and a large range of chaos(see
Fig.5c, the amplification of
Fig.5b). This shows
m0 changes the long term behavior greatly.
How the initial population size, x0 and y0, affects the long term dynamics of the two populations?
All the above simulations use the same initial population size:x0=0.2 and
y0=0.3, what if this condition changes? We discuss it briefly.
The method to plot Fig.4 is employed here. Other parameters are set as below:
m0=0.6
r=3.50
Thus we can know how initial population size changes the fluctuating period (see
Fig.6). This is a fractal. The range
(0,10-4) has been amplified by 10000 but similar details also exists. In fact, when I amplify the region by 10¹² I still find the same pattern. This shows a very little change in initial conditions will result different long term dynamics. For example, when
y0=0.00003,x0=0.00002, population x show 2-period cycling; but if change
x0 to 0.00003 (the difference is 0.00001), population x will show 4-period cycling.
However, such sensitivity is not universal for the entire plane. It exists in the area near (0,0),(0.3,0.3),(0.7,0.7) and (1,1). Other areas, especially far from the diagonal, don't show such sensitivity.
Conclusion
The long term dynamics of such a two-linked-population is sensitive to the initial conditions and the critical emigration value in some ranges.