3. The second principle, cooperation
Alan Berryman
Thursday, 07 May 2009 22:16 UTC
For those who are unfamiliar with this principle, and the others that I will cover later, I suggest they read about it in my book (Berryman 1999). I outline the basic ideas below.
For the moment we will consider that R, the per-capita rate of change over a time interval t-1 to t, is expressed as a function of population density, and that all other factors are assumed, for simplicity, as constants; I call this the R-function. Knowing that R is a function of the number of individuals at the beginning of the time interval, we can then calculate the number of individuals at the end of the time interval by employing the first principle.
When the second principle is in operation, cooperative interactions between organisms in a population dominate the R-function; i.e., social behavior like group hunting or group defense, and just the random process of being in a crowd that provides an individual with an advantage. In this sense, R rises directly with population density until it reaches a maximum rate at the maximum fecundity minus the minimum mortality rate of the population. If the R-function crosses R=0 then an equilibrium is formed which is unstable, because when R is above it the population grows and when R is below it goes extinct. Thus, equilibria caused by positive feedback are often called unstable thresholds because they separate the dynamics into regions of growth and collapse.
What this means is that populations diverge from the equilibrium points and, therefore, do not remain near to them for long. For this reason, most of the ideas about cooperative processes must be formulated by reason rather than fact, and this is probably why they have not received the attention they deserve. Most of the popular textbooks often completely ignore cooperation or deal with it peripherally. For example, Turchin (2001) does not list cooperation amongst his “laws” of population ecology, nor does he consider it an important component of Complex Population Dynamics (Turchin 2003), even though unstable cooperative thresholds produce the most complex population behavior possible (as we will see later). Royama (1992) also has little to say about cooperation in his landmark book Analytical Population Dynamics. It is strange why this principle is so widely neglected by ecologists when it is obviously essential to any comprehensive theory of ecological dynamics.
We can define the operation of the process as follows: Cooperation operates whenever interactions between individuals, whether purposeful or accidental, give rise to a direct (positive) effect of population density on the average per-capita rate of change of the species, up to some maximum value determined by its genetics and environmental conditions.
Berryman, A. A. 1999. Principles of population dynamics and their application. Stanley Thornes (Garland Science, Taylor & Francis).
Royama, T. 1992. Analytical Population Dynamics. Chapman & Hall.
Turchin, P. 2001. Does population ecology have general laws? Oikos 94: 17-26.
Turchin, P. 2003. Complex Population Dynamics: A Theoretical/Empirical Synthesis. Princeton University Press.
Updated 30 May 2009 15:39 UTC
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Replies
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This interests me because I have for some time been writing about depensation, the Allee effect, etc. in the fisheries context, because it has been almost completely ignored in the fisheries management world but is obviously relevant with respect to the dangers of accidental extinctions. For many years those who considered it at all tended to assume, in their statistical analyses, that the appropriate null hypothesis is that there is no “cooperation” manifest at low population levels, and mostly found no statistically significant evidence. But when Worm posed the opposite null, lo and behold nearly a third of the cases – data sets – examined indicated possible depensation. Sidney Holt.
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I have never understood why the Allee effect has gotten so much attention – for most cases I think it is highly speculative that positive effects of cooperation is really a function of population abundance on the scale of population dynamics, although I am not questioning that this could in principle be the cases. Also is there any really good evidence of this?
Instead I think the Allee effect is a speculative consequence of the wrong population dynamic theory in the first place: With traditional density regulation the growth rate will only decline with density – hence to have something that might explain the cases in nature were we see the opposite – the Allee effect was invented.
But as I argue elsewhere (e.g., under the first principle) traditional population dynamic theory (including exponential growth and traditional density regulated dynamics) is only a special case that relate to cases where there are no heritable variation between the individuals in the population. This, however, is not true for natural populations, and with heritable variation present – traditional density regulated dynamics evolves into inertial dynamics, where one of the basic expectations is that the growth rate will often increase in populations with small densities. Hence, with heritable variation in the population, we no longer need the Allee effect to explain that the growth rate can increase in small populations.
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