Theoretical Ecology:

| General Description | Complementation | Complexity |

Complementation in a simple two-linked-population model

 

The long range dynamics of the density of a single population has been well studied since Robert R. May’s pioneering work .1-7 Such researches give many surprising outcomes such as bifurcation, chaos, period-doubling reversals7, etc. When a system consisting of two linked populations is considered, more complex phenomena occur, one of which is complementation.  Here we show complementation is of much help for forecast in time and in place.

 

Model and analysis

A system consisting of two synchronized, homospecies populations X and Y, linked by migration, is considered.8,11  For each of the populations, the growth can be divided into two phases8: (1) intrinsic growth without migration; (2) migration between the populations.  The density after phase (1) and before phase (2) is examined.

To simplify the problem, we make the following assumptions: (1) No other population is linked to the above ones. The emigration of one population is the immigration of the other; (2) The environments of the two populations are identical except for geological sites and so the two populations share the same intrinsic growth rate and the emigration critical point (see below).

The following difference equations are employed to describe the dynamics of the two populations’ densities based on the above assumptions:

Xn+1 = f[Xn – m(Xn) + m(Yn)]

Yn+1 = f[Yn – m(Yn) + m(Xn)]                                         (1)

where

f(z) = r·z·(1 - z)                                              (2)

m(z) =    z– m0   z≥m0

           0      z<m0                                         (3)

The relationship between the density of the nth generation and that of the (n+1)th is described in equation (1), where emigration is represented by function m(X), which is determined by equation (3).  The most commonly used equation, the logistic equation — equation (2), is employed to describe intrinsic growth9,10.  The parameter r represents the intrinsic growth rate.

The reason for assigning such a format to m(X) is that it is more practical than just a constant, which is employed as the migration rate in many models2,4,7.  Emigration only occurs when the density of the population reaches a certain value — m0, which we call the emigration critical point.  When the population density is greater than this point the emigration will increase following the rising in the population’s density.

 

Complementation

To investigate the dependence of the long range dynamics on the intrinsic growth rate, the densities of the two populations after an enough number of generations were computed to a series of r7.  For each of 600 values of r, equation (1) was iterated 200 times and the last 100 points of each population were plotted in different colours in Fig 1b. The initial population density of X (X0) is 0.2 and Y (Y0) 0.3. The emigration critical point (m0) is 0.6. For comparison, such dependence in an isolated population (population T ), i.e. without migration, is plotted in Fig 1a.  The growth of population T is described by the logistic equation:

Tn = r·Tn·(1 - Tn )                                                         (4)

In Fig 1 we can see that the dynamics of the two linked populations’ densities is much more complicated than that of an isolated one.  The most notable phenomenon is that although Fig 1b shows almost the same bifurcating as that in Fig 1a, in fact each population occupies only one branch in a certain region.  We call this new phenomenon complementation and the region the complementary region, marked by C in Fig 1b.

Populations X and Y share the same dynamics — period 2, which is also the same as that of an isolated one, where r is less than 3.44 and greater than 3.30.  When r is greater than 3.44, population T shows period doubling with the increase of r as shown in Fig 1a. However, as shown in Fig 1b, population X and Y do not double their periods at r = 3.44 when migration links them.  To a certain r in the complementary region, the period number of each population is the half of that of an isolated one.

Strange phenomenon appears when r increases to a certain value (3.593): complementation disappears suddenly. We call this value the complementary critical point, represented by P in Fig 1b.  Each population suddenly occupies the other’s branch besides its own one where r is greater than the complementary critical point.  We call the region where r is greater than the complementary critical point the noncomplementary region, marked by U in Fig 1b.

I also analyzed the Ricker model and the Hassell model commonly used in ecological models5,7,9:

f(z) = z·e r (1 - z)                                                 (5)

f(z) = r·z·(1 + z)-4                                             (6)

The analysis reveals that a system consisting of two populations linked by migration often shows complementation.  So complementation is a common phenomenon.  One possible explanation for such phenomenon is that, contrary to the intrinsic growth rate, the increase of which will perturb the population and make its density fluctuate, migration would lessen the surviving pressure and augments the population stability.  In complementary region fluctuating extent is only the half of that in noncomplementary region and that in isolated populations (Fig 1).  After the complementary critical point, the effect of intrinsic growth rate on population is much greater than that of migration, so complementation disappears suddenly and fluctuating extent almost doubles.

 

Complementation and forecast

Complementation is of much help for forecast in time and place, which is one of the major aims of theoretical ecologists’ research, because complementation reduces density fluctuating extent, and so reduces uncertainty.  Here we give uncertainty such a definition as this: a range of values, not a point, of the to-be-forecasted quantity corresponds to a fixed value of the known quantity. To see how forecast becomes more precise in the complementary region, uncertainty in the complementary region and in the noncomplementary region is compared in Fig 2 and 3. Both in Fig 2 and 3, equation (1) was iterated 5500 times and the last 5000 points were plotted. The initial population density of X is 0.20 and Y 0.30.  The emigration critical point is 0.60.  For diagram a, r = L, for b, r = R. Fig 2 shows how complementation will be helpful for forecast in time and Fig 3, in place.  As seen in Fig 2, to a fixed density of current generation, although the next generation’s density is uncertain both in the complementary region and in the noncomplementary region, the latter is much more uncertain since its uncertain extent is much greater. So complementation augments the precision of forecast in time, i.e. forecast the next generation’s density with the aid of the current generation’s density.  Fig 3 shows complementation augments the precision of forecast in place, i.e. forecast the density of the one population(say Y) with the help of the other population’s density(say X).

 

Complementation shows how a system (say, a population) reduces its complexity and makes itself easier to forecast by linking with other systems, and so shows its significance. But is far from sufficient because in nature, generally more than two populations are linked and they form a network.  What is the dynamics of such a network is to be studied.

 

Reference:

1.May, R.M. Simple mathematical model with very complicated dynamics. Nature 261,459-467(1976).

2.Rohani, P. & Miramontes, O. Immigration and the persistence of chaos in population models. J. THEOR. BIOL 175(2),203-206(1995) .

3.Vandermeer, J. Period 'bubbling' in simple ecological models: Pattern and chaos formation in a quartic model. ECOL. MODEL. 95(2-3),311-317 (1996).

4.Sinha, S. Are ecological systems chaotic? An enquiry into population growth models. Current-Science-Bangalore 73 (11), 949-956(1997) .

5.May, R.M. Theoretical Ecology(Blackwell Scientific Publications,Oxford,1976).

6.Zimmer, C. Life After Chaos. Science 284,83-86,(1999).

7.Stone, L. Period-doubling Reversals and Chaos in Simple Ecological Models. Nature 365(14),617-620(1993).

8.Ruxton, G. D. Synchronisation between individuals and the dynamics of linked populations. J. THEOR. BIOL 183(1), 47-54(1996).

9.Lloyd, A.L. The coupled logistic map: A simple model for the effect of spatial heterogeneity on population dynamics. J. THEOR. BIOL 173(3), 217-230(1995).

10.Hastings, A. Complex interactions between dispersal and dynamics:lessons from coupled logistic equations. Ecology 74(5),1362-1372(1993).

11.Parthasarathy, S. & Guemez, J. Synchronisation of chaotic metapopulations in a cascade of coupled logistic map models. ECOL. MODEL. 106,17-25(1998).

 

Acknowledgements. I thank Professor Liu Shida for his help.