3 Single species models. Reading: Otto & Day (2007) section 4.2.1

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1 3 Single species models 3.1 Exponential growth Reading: Otto & Day (2007) section We can solve equation 17 to find the population at time t given a starting population N(0) = N 0 as follows. N(t) = (1 + b d)n(t 1) = (1 + b d)(1 + b d)n(t 2) = (1 + b d) 2 N(t 2) =... = (1 + b d) t N 0 From our model we can now derive the most basic result about births and deaths. If b > d then the population grows, i.e. (1 + b d) t as t. Conversely, if b < d then the population declines, i.e. (1 + b d) t 0 as t. If b = d then the population remains constant. 60

2 Malthus equation (equation 18) dy dt = (b d)y can be solved by remembering that the exponential function is its own derivative (see section 1.3.7). Guess the solution and check it! y = exp((b d)t) More explicitly we can solve it as follows N(t) N 0 1 N dn = t (b d)dt 0 ln(n/n 0 ) = (b d)t N(t) = N 0 e (b d)t Thus the population will grow or decrease exponentially. 61

3 3.2 Logistic growth Reading: Otto & Day (2007), sections and 6.6 Although the logistic growth equation (equation 19) is non-linear, it is solvable by separation of variables N(t) 1 N 0 N(1 N/K) dn = t rdt 0 N(t) 1 N 0 N dn + N(t) 1 dn = rt N 0 K N Integrating gives N 0 Ke rt N(t) = K + N 0 (e rt 1) This is exactly the sigmoidal equation

4 This tells us that, provided r > 0, the population will grow exponentially at first (since when t is small, K >> N 0 (e rt 1)) and will stabilise at a population of K (since N(t) K as t ). We illustrate this below. 63

5 By plotting f(n), as below, we can get an intuitive understanding of the behaviour of equation 19 without doing any integration. This figure also shows us that a population of K is globally stable, in the sense that all initial conditions converge to K. We will now formalise this concept of stability. 64

6 3.3 Steady states, stability and bifurcation diagrams Reading: Otto & Day (2007), primer P1.3, sections Unfortunately, the two models from the previous section are just about the only two models of population dynamics that are completely solvable, in the sense we can find a simple expression for N(t) in terms of t. We now introduce a number of concepts that will allow us to look at other, more complex, single species population models and understand their behaviour without solving them explicitly. Definition: The steady states of dn dt f(n ) = 0. = f(n) are the constants N such that Definition: A steady state N is said to be locally stable if every initial state close enough to N continues to be permanently close to N. 65

7 This last definition isn t precise enough for us to be able to do calculations about stability. In order to make it more mathematical we will now introduce ways of approximating functions close to steady states. Recall from the definition of a derivate in section that for small x we can approximate f f(a + x) f(a) (a) x Re-arranging we get f(a + x) f(a) + xf (a) (29) This corresponds to the idea that locally, a function can be approximated by a straight line. We can use this to understand various simplifying approximations for small x exp(x) = ln(1 + x) = x = 66

8 Equation 29 is part of a more general result known as Taylor s theorem. It says that most functions f(x] can be written as a power series expansion around a, i.e. f(x) = f(a) + (x a)f (a) + (x a)2 f (a) ! When x is near to a then (x a) is small and (x a) 2 is even smaller. This gives further justification to the linear approximation f(x) = f(a) + (x a)f (a) + higher order terms(h.o.t) or f(x) f(a) + (x a)f (a) 67

9 Definition: A steady state N is linearly stable if f (N ) = df dn N=N < 0 while N is said to be linearly unstable if f (N ) > 0 and neutrally stable if f (N ) = 0. Using Taylor s theorem we can now motivate this defintion. If we set N(t) = N + n(t) where n(t) << 1 then substituting into f(n) we get (using Taylor s theorem) dn dt = f(n + n) = f(n ) + f (N )n + higher order terms f (N )n Thus, as with the solution to equation 18 n(t) n 0 e f (N )t We call n 0 a small perturbation. If f (N ) > 0 the perturbation will grow, while if f (N ) < 0 the perturbation will shrink. Hence, the above definition of linear stability. 68

10 Exponential growth, equation 18, has only one steady state, N = 0. This is (linearly) stable if d > b and unstable if b > d. Logistic growth, equation 19, has two steady states, N = 0 and N = K. f (0) = r and f (K) = r. Thus if r > 0, 0 is unstable and N is stable. 69

11 Definition: A bifurcation diagram of dn dt = f b (N) for parameter b is a plot of the position and stability of the steady states of f b (N) as a function of b. Below we plot the bifurcation diagram for equation 19 as a function of K. 70

12 4 Population dynamics: fishing policy 4.1 Harvesting a population with logistic growth Reading: Otto & Day (2007), section 3.6. In fishing, we want to find the maximum sustainable yield (MSY) that we can take from the population. For example, consider a population that when not fished exhibits logistic growth according to equation 2. We can add a death term to the fish population where each fish is caught at a constant rate E: dn = f(n) = rn(1 N/K) EN (30) dt where E is the effort put in to catching fish by the fishermen. We define Y (E) = EN to be the fishermen s yield. It is this they aim to maximise. The steady states of 30 are N = 0 and N = K(1 E/r), provided that E < r. The yield is Y (E) = EN = EK(1 E/r) To find the MSY we differentiate with respect to E and find that E M 71 = r/2 is

13 maximal. This gives yield Y M (r/2) = rk/4 This MSY gives a stable fish population, since N = K/2 and f (N ) = r 2rN /K r/2 = r/2 = E M r < 0 We should contrast the MSY with the maximum short term yield, which is simply to take all the fish away from the unfished population. This would give a short term yield of K but leave no fish! 72

14 4.2 Crashes in fish stock Reading: Otto & Day (2007), section Consider the following model for a fish population N: dn dt = rn(1 N/K) EN b(n) where r, K and E are as before and b(n) is predation by seals. One possible form for b(n) is as follows b(n) = BN 2 A 2 + N 2 where B and A are constants. This is a Hill function (equation 12) with n = 2. 73

15 This form of predation is known as sublinear. For low densities the seals do not focus on the fish, but as the fish population increases the seals will focus on them more and more on catching them. We will return to predation functions in section 9 when we look at predator-prey models. The full model for the fish population is then 2 dn BN = f(n) = rn(1 N/K) EN (31) dt A 2 + N 2 This model is much more complex than those we have looked at previously. It has five parameters (with units): r (time 1 ), K (biomass), E (time 1 ), A (biomass) and B (biomass.time 1 ). To simplify the model we can non-dimensionalise it. Set, Note that, Equation 31 becomes ˆN T du dt = (r E) ˆNu u = N/ ˆN and τ = t/t d dt = d dτ dτ dt = 1 T 1 d dτ r ˆNu B ˆN 2 u 2 K(r E) A 2 + ˆN 2 u 2 74

16 du dt = (r E)T u 1 r ˆNu K(r E) B(T/ ˆN)u 2 (A 2 / ˆN 2 ) + u 2 Now comes the trick. We can choose ˆN and T, which are dimensionless quantities, to be any values we like. This choice can be motivated by some biological knowledge, or can be quite simply aimed at simplifying the model. Here we simplify the model by choosing ˆN = A, T = A/B. Giving du dt = su(1 u/q) u2 1 + u 2 (32) where (r E)A K(r E) s = and q = B ra It is best at this point to check that our non-dimensionalisation is correct! 75

17 We can now find the steady states of our simplified model, equation 32 u = 0 or s(1 u /q) = This last equation is best studied graphically. u 1 + u 2 76

18 Still without solving equation 32 we can also determine the stability of the three steady states graphically. 77

19 Finally, we now have enough information to sketch a bifurcation diagram for parameters q and s. We do this below. The bifurcation diagrams tell us something very important about the effects of fishing: either reducing s or q can result in a sudden crash in the fish population. In terms of our original parameters, an increase in E (fishing effort) corresponds to a decrease in both s and q. Thus a small increase in fishing effort can result in a large drop in fish populations. Furthermore, the reduction in effort required to recover the fish population must be much larger than the increase that caused the crash. 78