1.Single Population without Migration

In the beginning I studied the long-range dynamics of the density of an isolated population. The results my computer showed always made me surprised and excited. In the research on the dependence of such dynamics on intrinsic rate, I found period bifurcation and chaos. Bifurcation and chaos always occur when I choose several functions as growth model, such as Logistic model, Hassel model and Ricker model, etc. Why such functions show the same bifurcating phenomena still remains unanswered.(See Fig 1,2,3)

Table 1. Caption to figure 1-5

Xn+1 = f(Xn)

Model	f(X)	Figure
1.Logistic	r·X·(1-X)	Fig 1
2.Hassel	X·e^r(1-X)	Fig 2
3.Ricker	r·X·(1+X)^-4	Fig 3
4.	r·sin(X)	Fig 4
5.	r·X X<0.5 r·(1-X) X>0.5	Fig 5

r: intrinsic growth rate

Models 4 and 5 are not used in ecology modeling, but Model 4 is used in heart-beating model, and Model 5 in mathematical analysis.(Fig4,5)

To find the answers I read some books such as Robert May's Theoretical Ecology. I got to know bifurcating and chaos is a common phenomenon in isolated population systems. This means population density shows intrinsic fluctuating and nature does not show its balance. Of course it is still in debate.

2.Single Population with Migration

When considering immigration, a new phenomenon occurs - period doubling reversal. A very small immigration rate may easily prohibit chaos. With the increase of the intrinsic growth rate, the period number changes from 1 to 2 to 4...2^n and then to 2^(n-1) to...to 8 to 4 to 2. I never imagine such a small immigration rate will lead such dramatic results. As I think, immigration will increase survival pressure and make population unbalanced. However, the result shows immigration increase the balance. It is really a puzzle. When I told my chaos teacher such a phenomenon, he suggested me to write a paper. Then I searched and read many journals on ecology as my references, however, such phenomenon had been reported in Nature, 1993.

Period doubling reversal shows important consequence: it gives one way to inhibit chaos. Also, it shows significance in mathematics: it shows how chaos occurs.(see from Fig 6f to 6a)

Of course using just a constant to represent immigration rate is a too simple way, when I chose a density-dependent immigration rate, it also shows period-doubling reversals.

Table 2. Caption to figure 6

Xn+1 = f(Xn)

Model

f(X)

Figure

X·e^r(1-X) +m

m:immigration rate

Fig 6a~6f

3.Two Populations Linked by Migration

Then I considered a system consisting of two populations linked by migration. It showed new phenomenon such as sensitivity to population's initial density, jumping and complementation, etc. A single population's dynamics does not show sensitivity to initial density, but two populations do. (see the paper Complexity in a simple two-linked-population model)I found no papers describing jumping and complementation, so I wrote a paper on complementation. (see the paper Complementation in a simple two-linked-population model)

I also studied a system consisting of three populations, it shows more complex phenomenon.