1. Community Ecology BCB321 Mark J Gibbons, Room 4.102, BCB Department, UWC Tel: 021 959 2475. Email: mgibbons@uwc.ac.za Image acknowledgements – http://www.google.com

2. Not always an obvious relationship between predator and prey Sometimes….. Southern (1970) J Zoology 162: 197-285 Dempster & Lakhani (1979) J Animal ecology 48: 143-164

3. Modeling Predator Prey Dynamics Lotka-Volterra For populations displaying continuous breeding d N d t = r.N N t+1 - N t t 1 – t 0 = 1 Intrinsic rate of natural increase Exponential Growth Prey individuals are removed at a rate that depends on the frequency that predators encounter prey. Encounters increase with predator (P) and prey (N) numbers. The exact numbers encountered and actually consumed will depend on the searching and attack efficiency of the predator (a) – aPN N = number of prey d N d t = r.N – a.P.N N t+1 = N t + r.N t – a.P t.N t

4. d N d t = r.N – a.P.N In the absence of prey, predator numbers will decline exponentially through starvation d P d t = – q.P Where q = predator mortality rate Counteracted by predator birth – assumed to depend only on the rate at which food is consumed (a.P.N) and the efficiency (f) at which a predator converts this to offspring d P d t = f.a.P.N – q.P Lotka-Volterra equations * * P t+1 = P t + f.a.P t.N t – q.P t N t+1 = N t + r.N t – a.P t.N t COPY THESE EQUATIONS

5. N t+1 = N t + r.N t – a.P t.N t IF dN dt = 0 Equilibrium Solutions What is the population size of the predators that induces no change in the prey population size dN dt = 0 0 = r.N t – a.P t.N t Then r.N t = a.P t.N t Dividing by N t r = a.P t r/a = P t Dividing by a dN dt = r.N t – a.P t.N t dN = N t+1 - N t NOTE –This is a constant. Prey population size at EQUILIBRIUM not determined by this solution – population size can be stable at any size as long as predator population at specified size

6. P t+1 = P t - q.P t + a.f.P t.N t IF dP dt = 0 Equilibrium Solutions What is the population size of the prey that induces no change in the predator population size dP dt = 0 0 = -q.P t + a.f.P t.N t Then q.P t = a.f.P t.N t Dividing by P t q = a.f.N t q/a.f = N t Dividing by a.f dP dt = -q.P t + a.f.P t.N t dP = P t+1 - P t NOTE – This is a constant. Predator population size at EQUILIBRIUM not determined by this solution – population size can be stable at any size as long as prey population at specified size

7. Prey Population Predator Population r/a = P t dN dt = 0 When r/a Prey populations will increase in size at predator densities less than r/a, but will decrease in size at predator densities greater than r/a

8. Prey Population Predator Population dP dt = 0 When q/a.f = N t q/a.f Predator populations will increase in size at prey densities greater than q/a.f, but will decrease in size at prey densities less than q/a.f

9. Prey Population Predator Population r/a q/a.f Combining the two equilibria Prey Population Predator Population r/a q/a.f

10. Open a spreadsheet in MSExcel How do different values of r, a, q, N 0, P 0, and f influence the outcomes of species interactions? Next – Project a prey and a predator population into the future for 100 time units – using these two equations P t+1 = P t + f.a.P t.N t – q.P t N t+1 = N t + r.N t – a.P t.N t BUT……….. MUST constrain BOTH populations so that if numbers drop to zero, they remain at zero. Use an IF argument =IF(G2+(B$4*G2)-(D$4*H2*G2)>0, (G2+(B$4*G2)-(D$4*H2*G2),0)

11. Plot the two populations on a line graph It should look something like this: Coupled Oscillations Plot predator and prey numbers on an X-Y graph It should look something like this: Coupled Oscillations

12. THE END Image acknowledgements – http://www.google.com