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Unformatted text preview: 8 Ecosystem stability References: May [47], Strogatz [48]. In these lectures we consider models of populations, with an emphasis on the conditions for stability and instability. 8.1 Dynamics of a single population Set N = population size r = growth rate 8.1.1 Exponential growth The simplest possible growth model has N merely growing at rate r : d N = rN, r > . d t The solution predicts exponential growth : N ( t ) = N (0) e rt . Such a model works works well when the resource that a population requires for growth is suﬃciently abundant. In cases of such unlimited growth, 1 d N r = = ‘per capita’ or ‘specific’ growth rate N d t is constant, independent of N . 8.1.2 Logistic growth Eventually, however, resources become depleted and/or the population be comes overcrowded. 156 Thus as N increases, we expect that r will decrease. Moreover it can even become negative, meaning that the death rate exceeds the birth rate. Graphically, we expect that r ( N ) behaves like The point at which r ( N ) = is a special population size which neither grows nor decays. It corresponds to N = K , where K = carrying capacity . In most cases we can’t really know the shape of r ( N ), but the notion of a carrying capacity is itself reasonably sharp. So the simplest assumption would be to assume that r ( N ) decreases linearly: Then N r ( N ) = r 1 − K and our growth model now reads d N N d t = rN 1 − K , (56) known as the logistic equation or the Verhulst model . The logistic equation can be solved exactly but our goals are better served by analyzing it qualitatively. First, we plot d N/ d t vs. N , i.e., we plot the RHS of the logistic equation: 157 We consider only N ≥ since populations cannot be negative. Note that N ˙ = 0 , N = 0 or K ˙ N > , < N < K ˙ N < , N > K We call N = and N = K fixed points because they correspond to neither growth nor decay. We denote them with asterisks: N 1 ∗ = 0 , N 2 ∗ = K. Between N 1 ∗ = and N 2 ∗ = K , N grows toward K ; conversely, for N > K , N decays toward K . Thus we say that N 1 ∗ = 0 is unstable and that N 2 ∗ = K is stable . Indeed, we see that as long as we initiate growth with N (0) > , the popu lation always evolves to the carrying capacity. How it evolves depends on the initial condition: • Small populations N < K grow exponentially and later slowly towards K , sometimes called Sshaped or sigmoidshaped growth. • Large populations N > K decay exponentially to K . 158 Although specific details of the logistic model should not be considered as truly representative of population growth, the basic picture of exponential growth followed by saturation is widely but not universally observed. 8.1.3 Hyperbolic growth References: von Foerster et al. [49], Cohen [50], Johansen and Sornette [51]....
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 Fall '11
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