Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Modeling the Predator-Prey Relationship Michael Olinick Middlebury College MAA Session on Environmental Mathematics January 12, 2006 A basic assumption of the classic LotkaVolterra predator-prey model is that each species experiences exponential growth or decay in the absence of the other, Recent extensions of this model investigate logistic growth of one species when the other is absent, time delays in response by one species to population changes in the other, and multiple species interactions. A survey of consequences and predictions of the original and the modified models will be presented. “Nature offers a puzzling variety of interactions between species.” – Nino Boccara, Modeling Complex Systems Mutualism Competition Predation Predation Parasites: live a major period of their life in a single host doing nonlethal harm to the host (tapeworm) Grazers: attack large numbers of prey during their lifetimes, consuming part of the host without causing death ( mosquitos feeding on human blood, sheep grazing on grass). True Predators: attack many prey during their lifetimes, quickly killing the prey (wolves). Googling “Predator-Prey” returns 609,000 hits. Today: A very brief look at a very large and active area of modeling • • Review of the classic Lotka-Volterra: assumptions, model formulation, analysis and predictions, successes and failures. • Modifications of the Lotka-Volterra Approach. A Real World Puzzle Umberto D’Ancona asks Vito Volterra: After World War I, the proportion of predatory fish caught in the Upper Adriatic was considerably higher than in the years before the war; the proportion of prey fish was lower. The war between Austria and Italy halted commercial fishing. How did the war benefit the predators more than the prey? Alfred James Lotka 1880-1949 Vito Volterra (1860-1940 Elements of Physical Biology, Baltimore: Williams and Wilkins, 1925. (Dover reprint Elements of Mathematical Biology, 1956) “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi,” Rendiconti dell’Academia dei Lincei, 6 (2), 31 – 113 (1926). (“Fluctuations in the Abundance of a Species Considered Mathematically,” Nature, 118, 558 – 560 (1926). Francesco M. Scudo and James R. Ziegler, eds., The Golden Age of Theoretical Ecology, 1923 – 1940, Berlin: SpringerVerlag, 1978. The Lotka-Volterra Model Assumptions A classic formulation of this situation is that of a population of rabbits, who feed on clover, and a population of foxes, who feed on rabbits. The assumptions usually made about the situation are these: 1. In isolation, the rate of change of population of one species is proportional to the population of that species. In the absence of foxes, it is assumed that the rabbits population will exhibit exponential growth. If there are no rabbits, the fox population will undergo a pure death process. 2. There is always so much clover that the rabbits have an ample supply of food. The only food available to the predatory foxes are the rabbits. 3. The number of kills of rabbits by foxes is proportional to the frequency of encounters between the two species. This, in turn, is proportional to the product of the populations of rabbits and foxes. Thus there will be few kills if there are few rabbits or few foxes and many kills only when both populations are relatively large. The Lotka-Volterra Model Mathematical Formulation x = measure of prey (rabbit) population (population, biomass) y = measure of predator (fox) population (population, biomass) dx = ax bxy dt dy = mxy ny dt dx = ax bxy = x(a by) dt dy = mxy ny = y(mx n) dt where a, b, m, n are positive constants. Can reduce the 4 parameters to 1: Let X = Then m a x, Y = y, T = an t, r = n b dX = rX (1 Y ) dT dY 1 = Y (X 1) dT r a n Analyzing the Phase-Plane for Lotka-Volterra Region dx dy dt dt ______________ I <0 >0 II <0 <0 III >0 <0 IV >0 >0 Signs of dx/dt and dy/dt for the Lotka-Volterra model. Elementary Observations: • There are two equilibrium points: (0,0) and (n/m, a/b). • The positive x-axis and the positive y-axis are each possible orbits. • Since L-V is system of autonomous differential equations, orbits starting in the first quadrant will remain there. The Volterra mapping technique can be used to find a more exact orbit to the original predator-prey model. Note that y (mx - n) dy = x (a - by) dx in the original system. After the variables are separated in this differential equation and integration is completed, the solution looks like a log y -by = mx -n log x + C which may be written as (ya e-by)(xm e-nx) = K. y = x m e nx Volterra Mapping Technique A typical result. Fig. 4.12 Orbits for the predator-prey system dxldt = ax -bxy, dyldt = mxy -ny obtained from the Volterra mapping technique. Here a = 5, b = 3, m = 2, and n = 6. Critical points are (0,0) and (3, 513). All orbits move in a counterclockwise direction. The constant K is determined by x0n y0a initial conditions (x0,y0) and is equal to by0+ mx0 At (3,5/3), K= .156. e Elliptical orbit of the linearized version of the predator-prey model. Volterra Solves the Puzzle After The War: dx = x(a by) dt dy = y(mx n) dt Average Prey Population is n m Average Predator Population is a b Before the War dx = x(a c by) dt dy = y(mx n d) dt n+d m ac Average Predator Population was b Average Prey Population was Linearizing dx = f (x, y) dt dy = g(x, y) dt If (x*,y*) is an equilibrium point, set u = x – x*, v = y – y*. In the neighborhood of this equilibrium point, neglecting quadratic and higher degree terms, we have f du f = (x*, y*)u + (x*, y*)v dt x y g dv g = (x*, y*)u + (x*, y*)v dt x y The general solution of the linear system is u (0) u (t) = exp(D f(x*)t)u where u (t) = (u(t), v(t)), f = (f , g), x * = (x*, y*) and D f(x*) is the Jacobian matrix of f at x* (often called the community matrix in ecology) f x (x*, y*) D f(x*) = g x (x*, y*) f (x*, y*) y g (x*, y*) y Lotka-Volterra Equilibrium Jacobian Eigenvalues a 0 0 n bn 0 m ma 0 b a,-n Point (0,0) n a m , b unstable neutrally i an , i an stable Does the Lotka-Volterra model predict patterns observed in measured real-world data? Changes in the abundance of the lynx and the snowshoe hare, as indicated in the number of pelts received by the Hudson’s Bay Company from 1845 to 1934. “This is a classic case of cyclic oscillation in population density.” BUT… In some cycles, the maximum of the lynx population led rather than lagged behind the hare population. The cycles in these two animals as observed in the Hudson Bay data are not described by predator–prey interactions of the Lotka–Volterra type. Michael Gilpin, “Do Hares Eat Lynx?” American Naturalist, Volume 107 (1973), 727 – 730. C. Modifying the Model Several variations of the Lotka- Volterra predator-prey model have been proposed that offer more realistic descriptions of the interactions of the populations. 1. If the population of rabbits is always much larger than the number of foxes, then the considerations that entered into the development of the logistic equation may come into play. If the number of rabbits becomes sufficiently great, then the rabbits may be interfering with each other in their quest for food and space. One way to describe this effect mathematically is to replace the original model by the more complicated system, dx x = ax(1 ) bxy dt k dy = mxy ny dt where the prey population grows logistically in the absence of predators, with a carrying capacity of k. Modified Lotka-Volterra (Carrying Capacity for Prey) Equilibrium Jacobian Eigenvalues Point (0,0) (k,0) n a m , b a 0 a,-n 0 n a bk ( a, mk n) 0 mk n an bn Complex with mk m negative real ma(mk n) 0 mkb part if k> n an + m 4m2 unstable Unstable If k > m/n Asymptot ically stable EXERCISE: Most predators feed on more than one type of food. If the foxes can survive on an alternative resource, although the presence of their natural prey (rabbits) favors growth, a possible alternative model is the system dx = ax -bx2 -cxy dt dy = mxy + ny - py2 dt where a, b, c, m, n, p are positive constants. 3. P. H. Leslie and J. C. Gower studied a third variation, the system of equations dx = ax - bxy dt dy y =(c - e dt x )y where the parameters a, b, c, e are again positive constants. Here the term y/x arises from the fact that this ratio ought to affect the growth of the predator. When foxes are numerous and rabbits are scarce, y/x is large and the growth of fox population will be small. Conversely, when the supply of rabbits is ample for the foxes, y/x is small and there is slight restriction on the increase of the predators. The orbits associated with the Leslie-Gower model are curves that spiral in toward the critical point. We illustrate a typical situation of this stable equilibrium. Results of the Leslie-Gower model for a predator-prey system. Here dx/dt = ax -bxy, dy/dt = (c -e(y/x) )y. The curves illustrated are for a = 1, b = .1, c = 1, e = 2.5 and initial populations x0 = 80, y0 = 20. The critical point is (25, 10). The top graph shows the orbit of a solution of the system of differential equations; it spirals in toward the critical point. The bottom graph shows x and y as functions of t. From E. Pielou, An Introduction to Mathematical Ecology, New York: John Wiley, 1969. Gary Harrison’s Models Each of the models we have seen makes the assumption of insatiable predators. Limits on gut size and time available for hunting dictate that the predators' kill rate will approach an upper bound as the density of prey increases. Gary Harrison, “Comparing Predator_Prey Models to Luckinbill’s Experiment with Didinium and Paramecium,” Ecology Volume 76 (1995), 357 – 374. dx x = ax(1 ) bf (x)y dt k dy = mf (x)y ny dt where f(x) is a measure of the functional response of the predator. Many predators exhibit a “saturation” effect in their functional response when prey is abundant. One way to model this is to use the function f (x) = x c+ x which is called a Holling Type 2 functional response. Leo S. Luckinbill, “Coexistence in Laboratory Populations of Paramecium Aurelia and its Predator Didinium Nasutum,” Ecology Volume 54 (1973), 1320 – 1327. The Model (aka Rosenzweig-MacArthur model) becomes dx x bxy = ax(1 ) dt k c+ x dy mxy = ny dt c + x M. L. Rosenzweig and R. H. MacArthur , “Graphical Representation and Stability Conditions of Predator-Prey Interactions,” American Naturalist, Volume 97 (1963), 209-223. Two trajectories in the (Prey,Predtaor)-plane converging to a stable limit cycle in Harrison’s model. Other Modifications/Approaches • Fluctuating Environments Carrying Capacity may vary with time. Replace k by k(t) Example: • k(t) = k(1+ qsin t) Discrete-Time Models The discrete version of dX = rX(1 Y ) dT dY 1 = Y (X 1) dT r is X(T + 1) = X(T ) + rX(t)(1 Y (T )) 1 Y (T + 1) = Y (T ) + Y (T )(X(T ) 1) r More generally, X(T + 1) = X(T ) + f (X(T ),Y (t)) Y (T + 1) = Y (T ) + g(X(t),Y (t)) In the neighborhood of the equilibrium point (x*,y*), neglecting quadratic and higher degree terms, we have f f (x*, y*)u(T ) + (x*, y*)v(T ) x y g g v(T + 1) = v(T ) + (x*, y*)u(T ) + (x*, y*)v(T ) y x u(T + 1) = u(T ) + General Solution of Linear System is u(T ) = ( I + Df (x*, y*)) u(0) T The equilibrium point (x*,y*) is asymptotically stable if the absolute values of the eigenvalues of I + Df(x*,y*) are less than 1. The discrete time analogue of Lotka-Volterra has the nontrivial equilibrium point (1,1). The eigenvalues of I + Df(1,1) are 1 + i and 1 –i. The neutrally stable equilibrium point of the continuous time model is unstable for the discrete time model. • Time-Delay Models K. Gopalsamy, Stability And Oscillations In Delay Differential Equations Of Population Dynamics, Kluwer Academic, Dordrecht/Norwell, MA, 1992. Yang Kuang Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993. • Lattice Models Nino Boccara, O. Roblin, and Morgan Roger, “An Automata Network Predator-Prey Model with Pursuit and Evasion,” Physical Review Volume 50 (1994), 4531 – 4541. • Multiple Species Models Werner Krebs, “A General Predator-Prey Model,” Mathematical and Computer Modeling of Dynamical Systems, Volume 9 (2003), 387 – 401.