Background

In natural ecological systems, a great variety of environmental influences, often of a stochastic nature, affect the development of communities over time. Such effects (for example due to climate change) may be so overwhelming that they force communities into a certain direction. But they may also be large or relatively minor, apparently random fluctuations (such as earthquakes or droughts), causing a great deal of uncertainty in the fate of populations and communities. Such effects are not discussed here. We restrict our discussion to the possibility or impossibility of predictions based on theoretical models.This discussion may also be of interest to people concerned with the apparent “randomness” vs. “lawfulness” in history, which will be discussed in another knol.

Unpredictability in single populations

May (e.g. 1974,1975 [1] [2]) and others after him, using the apparently simple nonlinear equation for population growth x_{t+1}= rx_{t }(1- x_{t}) (where x_{t}= size of population of generation t, x_{t+1}= size of population of generation t+1, r = intrinsic rate of population growth), have shown that chaotic ﬂuctuations in the size of populations arise due to high intrinsic (not affected by environmental influences) rates of population growth r. When population growth is low, populations are at equilibrium, reaching a steady-state with a single value of population size. When reproductive rates exceed 3, there are at ﬁrst 2 quasi-equilibrium values, followed (with increasing reproductive rates) by 4, 8 and 16 quasi-equilibrium points. At rates exceeding r=3.57, population sizes ﬂuctuate chaotically (Figure 1). These ﬂuctuations seem to be random, but are in fact strictly deterministic. The ﬂuctuations are highly sensitive to initial conditions, i.e. small differences in the initial population size lead to very large differences in future ﬂuctuations, which makes predictions about the future impossible.

Figure 1: Bifurcation diagram for a single population, showing population sizes x (as the proportion of the crowding level 1) plotted against reproductive rates r. Insets show changes of the population size plotted against time at three r-values Note: the population size is at a stable equilibrium with single values of x until a certain value of r (larger than 3) is reached. Beyond this, populations ﬁrst alternate between two quasi-equilibrium points, then between 4, 8 and 16. At r=3.57 ﬂuctuations in population size become chaotic, i.e. it is practically impossible to predict population sizes, although they are due to a strictly deterministic process. Note: x never exceeds the crowding level 1.

**We conclude**that predictions on population sizes to be expected for high reproductive rates are practically impossible for two reasons: fluctuations in population size depend on initial population sizes (even very minor differences may lead to vastly different outcomes), and fluctuations are so complex that, even if initial population sizes were known to an infinite degree of accuracy, exact predictions are impossible.

Unpredictability in multi-species systems

Simulation/animation methods

Population sizes in two-species systems can be described by the competitive Lotka-Volterra equations. Various parameters in the equations determine which species prevails, i.e., does not become extinct. In two-species systems, values of the competition factors alpha and beta indicate the effects which the presence of one species has on the population size of the other. Their maximum values are 1. Alpha indicates the effect species x has on species z, beta indicates the effect species z has on species x as a result of interspecific competition. Extinction times of the species depend, in addition to the values of the competition factors, on the reproductive rates r_{x}and r_{z}of species x and z, on the number of subpopulations within each species[3] and their reproductive rates, and on the initial population sizes x_{0}and z_{0}of the species. In three-species systems, a third species y with its parameters is used as well. Simulations based on these equations, in which all except one parameter are kept constant, and the single variable can assume many values, permits the stepwise evaluation of changing community patterns. Using an appropriate program, results can be “played” as a movie. Alternatively, each frame can be separately examined.Programs for two- and three-species competition animations can be found here.Important for the approach demonstrated here is that reproductive rates have a range, measuring the variability of rates within each species due to the presence of several subpopulations with different reproductive rates. This agrees with many natural systems: species often consist of a number of subpopulations.

Unpredictability in competing two-species systems

Examples of computer simulations illustrating the extinction of species at certain times (after a certain number of generations) are given in Figure 2. In Figure 2, we assume that initital population sizes both of species x and z are 0.5 (i.e., half the crowding level 1 which a species can ever reach). We further assume that competition factors and reproductive rates of both species covary (always have the same value), and that the reproductive rate of species z is always 3.7. In A, the reproductive rate of species x is 1.0, in B it is 2.73, and in C it is 3.57. – Extinction times (in numbers of generations) of species x is illustrated by hollow (Figure 2A left), of species z by grey, and of both species jointly by black. White illustrates that neither of the species becomes extinct.Figure 2A shows that species x becomes first extinct when ranges of reproductive rates do not exceed 0.3, and when competition factors are larger than 0.5. Neither of the species becomes extinct (white) when competition factors are smaller then 0.5 and ranges of reproductive rates smaller than 0.3. – Figure 2C shows that both species jointly or species z alone become almost immediately extinct. Only at very low values of the competition factors and ranges of reproductive rates is there no extinction of any species. – Note that predictions, even at one particular value of r_{z}, are practically impossible for certain parameter-values, for example at the border between hollow and white and hollow and gray in A. In animations using stepwise changing values of r_{x}, patterns in A,B and C change into each other and into many others not shown in the figure, corresponding to an even greater complexity and therefore unpredictability.

Figure 2. Simulations of two-species competition. Assumptions are as follows: initital population sizes both of species x and z are 0.5 (i.e., half the crowding level 1 which a species can ever reach); competition factors alpha and beta and reproductive rates r_{x}and r_{z }of both species covary (always have the same value); the reproductive rate of species z is always 3.7. In A, the reproductive rate of species x is 1.0, in B it is 2.73, and in C it is 3.57. – Extinction times (in numbers of generations) of species x is illustrated by hollow (A left), of species z by grey, and of both species jointly by black. White illustrates that neither of the species becomes extinct. Note that predictions, even at one particular value of r_{z}, are practically impossible for certain parameter-values, for example at the border between hollow and white and hollow and gray in A. In animations during which patterns in A,B and C change into each other corresponding to gradually changing values of r_{x}, complexity and therefore unpredictability are even greater. © Klaus Rohde

**We conclude**that, even under the simplified assumptions necessary to allow graphical representation, and even in two species-systems, predictions on which species will become extinct first and when, are very difficult and indeed practically impossible for many of the parameter values.

Unpredictability in competing three-species systems

Using Lotka-Volterra equations not for two, but for three competing species, simulations lead to even more complex patterns. We have run simulations (animations) in which all values except one were kept constant, and found radical changes in the extinction patterns even when changes in the value of the variable (for example reproductive rate of one of the three competing species) were relatively minor. When such changes were major, results were totally different. We show two examples. In Figure 3, the three species x, y and z have the following values: reproductive rates r_{x}=0.444, r_{y}=3.0, r_{z}=3.0, initial population sizes x_{0}=0.001, y_{0}=0.01, z_{0}=0.5. Extinction times (number of generations) for species x represented by green, for species y by red, for species z by blue, for species x+y by yellow, for species x+z by purple, for species y+z by light blue, for all species jointly by red-yellow. White indicates that neither species becomes extinct. Values for competition factors alpha, beta and gamma, and those for the ranges of reproductive rates, are assumed to covary (always have the same value). Each species consists of 5 subpopulations with different reproductive rates. – Note the very complex extinction pattern; only when r-values are larger and values of alpha, beta and gamma smaller than about 0.3 does no species become extinct (white). Predictions of extinctions are impossible for most parameter values because of the complexity of the pattern.

Figure 3. Three-species competition diagram. The three species x, y and z have the following values: reproductive rates r_{x}=0.444, r_{y}=3.0, r_{z}=3.0; initial population sizes x_{0}=0.001, y_{0}=0.01, z_{0}=0.5. Extinction times (number of generations) for species x represented by green, for species y by red, for species z by blue, for species x+y by yellow, for species x+z by purple, for species y+z by light blue, for all species by red-yellow. Note that values for competition factors alpha, beta and gamma, and those for the ranges of reproductive rates, are assumed to covary. Note the very complex extinction pattern; only when r-values are larger and values of alpha, beta and gamma smaller than about 0.3 does no species become extinct (white). Predictions of extinctions are impossible for most parameter values because of the complexity of the pattern. © Klaus Rohde

Figure 4. As Figure 3 but r_{x}=3.5556. Note that all species become extinct almost immediately when values of competition factors are greater than 0.5 (half the possible value). Neither species becomes extinct when competition factors are smaller than 0.2 and ranges of reproductive rates smaller than 0.4. The extinction pattern is very complex at intermediate values. Predictions of extinctions of those species that initially survived are impossible for most parameter values because of the complexity of the pattern. © Klaus Rohde In animations, in which r_{x}-values change in small steps, patterns in Figures 3 and 4 gradually change into each other and into many others not represented in Figures 3 and 4, leading to even greater complexity and therefore unpredictability.

**We conclude**that for most values of the various parameters, predictions of the outcome (winners of the competition) are practically impossible.

Unpredictability in cellular automata generated by very simple rulesWolfram’s (2002[4]) “A New Kind of Science” uses an approach to solving scientiﬁc problems which differs radically from those of traditional science. It has been made possible by high-speed computers. The idea on which NKS is based is to run a large number of computer programs based on a variety of “rules” and look at the results. Particularly useful has been the application of such rules to cellular automata (Wolfram 1986 [5], 2002 [4]). A cellular automaton is composed of rows of cells, each cell characterised by a particular state (such as black or white, or red and green). A rule speciﬁes how an automaton develops from one to the next computational state, based on its previous state and the state of its neighbours. Wolfram’s approach applied to evolution/ecology has been discussed in greater detail in another knol (see also [6]).Figure 5 illustrates the pattern generated by Wolfram’s rule 30 (if a cell and its right-hand neigbour were white in the previous step, make the new colour of the cell what the previous colour of its left-hand neighbour was. Otherwise, make the new colour the opposite to that). Some regularities can be recognized on the left of the figure, but below the initial middle cell (arrow) a seemingly random (=pseudorandom) row of black and white cells has been demonstrated by various statistical methods for more than one million steps, i.e. predictions of which type of cell occurs a certain number of steps down are impossible.

Figure 5. Cellular automaton generated with Wolfram’s rule 30 (i.e., if a cell and its right-hand neigbour were white in the previous step, make the new colour of the cell what the previous colour of its left-hand neighbour was. Otherwise, make the new colour the opposite to that). Left and right truncated. Note some regularities on the left (diagonal bands), and scattered throughout some triangles and other small structures, but overall the irregular appearance is overwhelming. The sequence below the initial black cell in the middle, over 1 million steps, has no repeats whatsoever, as revealed by various highly sophisticated statistical and mathematical tests, i.e. it is apparently random.

**We conclude**that even in cellular automata generated by very simple rules, predictions are often impossible.ConclusionThis brief discussion has shown that in some ecological models predictions of the outcome (population size in single-population simulations, extinction of species in multi-species systems) are often impossible, in spite of the fact that the models are strictly deterministic (i.e., lead to patterns that can be repeated in exactly the same way if exactly the same parameter-values are used).

References

May, R. M. 1974. Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186: 645-647.May, R.M. 1975. Deterministic models with chaotic dynamics. Nature 256, 165-166.Rohde, K. and Rohde, P.P. 2001. Fuzzy chaos: reduced chaos in the combined dynamics of several independently chaotic populations. American Naturalist, 158, 553-556.Wolfram, S. 2002. A new kind of science. Wolfram Media Inc. Champaign, Il. Wolfram, S. 1986. Theory and applications of cellular automata: Advanced series on complex systems. World Scientiﬁc Publishing: Singapore.Rohde, K. 2005. Cellular automata and ecology. Oikos,110, 203-207.————Copyright Klaus Rohde