Predator-Prey Dynamics for Rabbits, Trees, and Romance

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1 Predator-Prey Dynamcs for Rabbts, Trees, and Romance Julen Clnton Sprott Department of Physcs Unversty of Wsconsn - Madson sprott@physcs.wsc.edu.. Introducton The Lotka-Volterra equatons represent a smple nonlnear model for the dynamc nteracton between two bologcal speces n whch one speces (the predator) benefts at the expense of the other (the prey). Wth a change n sgns, the same model can apply to two speces that compete for resources or that symbotcally nteract. However, the model s not structurally stable, snce persstent tme-dependent (oscllatory) solutons occur for only a sngle value of the parameters. Ths paper consders structurally stable varants of the Lotka-Volterra equatons wth arbtrarly many speces solved on a homogeneous two-dmensonal grd wth couplng between neghborng cells. Interestng, bologcally-realstc, spato-temporal patterns are produced. These patterns emerge from random ntal condtons and thus exhbt self-organzaton. The extent to whch the patterns are self-organzed crtcal (spatally and temporally scale-nvarant) and chaotc (postve Lyapunov exponent) wll be examned. The same equatons, wthout the spatal nteractons, can be used to model romantc relatonshps between ndvduals. Dfferent romantc styles lead to dfferent dynamcs and ultmate fates. Love affars nvolvng more than two ndvduals can lead to chaos. The lkely fate of couples wth dfferent romantc styles wll be descrbed... Lotka-Volterra Equatons One varant of the Lotka-Volterra equatons (Murray 993) for two speces (such as rabbts and foxes) s dr / dt = r R( R a F) df / dt = r F( F a R) () where R s the number of rabbts and F s the number of foxes, both postve and each normalzed to ts respectve carryng capacty (the maxmum allowed n the absence

2 Predator-Prey Dynamcs for Rabbts, Trees, and Romance of the other), r and r are the respectve ntal growth rates n the absence of competton, and a and a determne the nterspeces nteracton. In a predator-prey model, the predator (foxes) would have r < 0 (snce they gradually de n the absence of rabbts to eat) and the other constants would be postve. However, the same equatons wth all postve constants could model competton, or wth both growth rates negatve could model cooperaton or symboss (chckens and eggs, plants and seeds, bees and flowers, etc.)..3. Equlbrum and Stablty The system n Eq. () has four equlbra, one wth no rabbts, one wth no foxes, one wth nether, and a coexstng one wth a R = a a a F = a a () whch s of prmary nterest. For the predator-prey case (a r > 0, a r < 0), the coexstng equlbrum s a stable focus for r ( a ) < r ( a ), at whch pont t undergoes a Hopf bfurcaton, after whch the trajectory sprals outward wthout bound from the unstable focus. Hence there are no structurally stable oscllatory solutons. For the competton case (a r > 0, a r > 0), the coexstng equlbrum s a stable node for a < and a <, at whch pont t undergoes a saddle-node bfurcaton, after whch one of the speces des whle the other goes to ts carryng capacty (R = or F = ). Thus there are no oscllatory solutons, and stablty requres that the ntraspeces competton domnates. When the nterspeces competton domnates, the weaker speces s extngushed by the prncple of compettve excluson or survval of the fttest (Gause 97). For the cooperaton case (a r < 0, a r < 0), both speces ether de or grow wthout bound n what May (98) calls an orgy of mutual benefacton. Wth N speces there are N equlbra, only one of whch represents coexstence, and t s unlkely that ths equlbrum s stable snce all of ts egenvalues must have only negatve real parts. Ecologcal systems exhbt dversty presumably because there are so many speces from whch to choose, because they are able to spread out over the landscape to mnmze competton, and because speces evolve to fll stable nches (Chesson 000). If many arbtrary speces are ntroduced nto a hghly nteractng envronment, most would probably de..4. Spato-temporal Generalzaton Now assume there are N speces wth populaton S for = to N and that they are spread out over a two-dmensonal landscape S (x, y). The speces could be plants or

3 Predator-Prey Dynamcs for Rabbts, Trees, and Romance 3 anmals or both. For convenence, take the landscape to be a square of sze L wth perodc boundary condtons (a torus), so that S (L, y) = S (0, y) and S (x,l) = S (x, 0). One commonly assumes that each speces obeys a reacton-dffuson equaton of the form S t N = r S ( a S ) + D S (3) j j= j Ths system s usually solved on a fnte spatal grd of cells each of sze d so that S (x, y) = [S (x+d, y) + S (x d, y) + S (x, y+d) + S (x, y d) 4S (x, y)] / d. Dffuson s perhaps not the best model for bology, however, and f too large, t tends to produce spatally homognety. Instead, assume that each speces nteracts not just wth the other speces n ts own cell but also n ts four nearest-neghbor cells (a von Neumann neghborhood), gvng S t N = r S ( S a S ) (4) j j=, j j where S j ( x, y) = S j (x+d, y) + S j (x d, y) + S j (x, y+d) + S j (x, y d) + α j S j (x, y) s a weghted average of the neghborhood. In the example of rabbts and foxes, you can thnk of α j as the tendency for the foxes to eat at home. Wth α j = 0, the foxes never eat at home, and wth α j = they forage unformly over a fve-cell neghborhood ncludng ther home cell. In the case of trees and seeds, ths term s where one would nclude a seed dsperson kernel. The example that follows uses α j = for all j, but the results are not senstve to the choce. Includng only nearest neghbor cells normalzes space so that the cell sze d s the order of the mean foragng (or dspersal) dstance. Note that tme can also be normalzed to one of the growth tmes, so that we can take r = wthout loss of generalty..5. Numercal Example In Eq. (4) all the bology s contaned n the vector r, the nteracton matrx a j, and the dspersal vector α j here taken as untary. Instead of modelng realstc bology, we choose the values of r and a j from an IID random normal dstrbuton wth zero mean and unt varance and examne many nstances of the model to explore a range of possble ecologes. For brevty, Fg. llustrates most of the common behavors. It starts wth sx speces wth unform random values S (x, y) n the range of 0 to 0. on a grd. The upper plots show the spatal structure of the sx speces after 00 growth tmes, and the lower plot shows the cumulatve relatve abundance of each speces versus tme. One speces (the fourth) des out. The frst, second, and sxth, nearly de,

4 4 Predator-Prey Dynamcs for Rabbts, Trees, and Romance and then recover, after whch the fve speces coexst wth aperodc temporal fluctuatons and spatal heterogenety. Fgure. Typcal example of a spato-temporal soluton Eq. (4) wth sx ntal speces, one of whch ded. Fgure shows the domnant speces n each cell after 00 growth tmes, each n a dfferent shade of gray. Ths dsplay facltates comparson wth real data and wth the results of cellular automata models (Sprott 00). As n earler studes, we defne the cluster probablty as the fracton of cells that are the same as ther four nearest neghbors and Fourer analyze the temporal fluctuatons n cluster probablty to obtan ts power spectrum. The result n Fg. 3 shows a power law over about a decade and a half, mplyng temporal scale nvarance as suggestve of self-organzed crtcalty (Bak 996). Other quanttes such as the total bomass ΣS (t) also have power-law spectra.

5 Predator-Prey Dynamcs for Rabbts, Trees, and Romance 5 Fgure. Landsccape pattern showng the domnant speces n each cell. Fgure 3. Power spectrum of fluctuatons n cluster probablty.

6 6 Predator-Prey Dynamcs for Rabbts, Trees, and Romance To assess whether the dynamcs are chaotc, we follow Lorenz (963) and round the values of S (x, y) to four sgnfcant dgts after the ntal transent has decayed and calculate the growth of the error n the total bomass as the perturbed and unpertubed systems evolve determnstcally. The result n Fg. 4 suggests an exponental growth n the error wth a growth rate the order of 0.r. If the system modeled a forest wth a typcal r of 50 years, the predctablty tme would be about 500 years. Fve speces s the mnmum number for whch such chaotc solutons were found. Wth four speces, lmt cycles are common, and wth three or fewer speces, all stable solutons attract to a tme-ndependent equlbrum wth no spatal structure. Spatal heterogenety always correlates wth temporal fluctuatons. Fgure 4. Exponental growth n total bomass error suggestng chaos. Note that the chaos and spatal structure arse from a purely determnstc model n whch the only randomness s n the ntal condton. In fact, smlar structures arse from hghly ordered ntal condtons, perturbed only by nose as small as one part n 0 6. The model s endogenous (no external effects), homogeneous (every cell s equvalent), and egaltaran (all speces obey the same equaton, wth only dfferent coeffcents). The spatal patterns and fluctuatons are nherent n the equatons whose solutons spontaneously break the mposed spatal symmetry. Smlar behavor has been observed n models wth a sngle spatal dmenson (Sprott, Wldenberg, and Azz 005).6. Applcaton to Romantc Relatonshps To stress the generalty of the model, we can apply t to romantc relatonshps. Imagne two lovers, Romeo and Julet, characterzed by a par of equatons such as Eq. () n whch R s Romeo s love for Julet and F s Julet s love for Romeo, both

7 Predator-Prey Dynamcs for Rabbts, Trees, and Romance 7 postve. Each lover can be characterzed by one of four romantc styles dependng on the sgns of r and a as shown n Table I usng names adapted from Strogatz (988). The varable r determnes whether one s love grows or des n the absence of a response from the other (a = 0), and the varable a determnes whether requted love enhances or suppresses the ntensty of one s feelngs. Table I. Romantc styles a Cautous Narcsscst lover nerd Hermt + Eager beaver r Wth two nteractng lovers, there are thus 4 = 6 dfferent combnatons of romantc styles, the fate of whch are determned by the strength of the nteratons. As an example, choosng r and a from an IID random normal dstrbuton wth zero mean and unt varance gves the results n Table II, where the percentages are the probablty that a stable steady state s reached. In some sense, the best parng s between an eager beaver and a narcssstc nerd, although two eager beavers have solutons that grow mutually wthout bound. Not surprsngly, the prospects are dsmal for a hermt and not much better for a cautous lover. Fortunately, humans often seem capable of adaptng ther romantc styles to ft the stuaton. Table II. Probablty of mutual stable love for varous parngs. Narcssstc Eager Cautous Hermt nerd beaver lover Narcssstc nerd 46% 67% 5% 0% Eager beaver 67% 39% 0% 0% Cautous lover 5% 0% 0% 0% Hermt 0% 0% 0% 0% It s also nstructve to examne love trangles, n whch case there are four varables f two of the lovers are unaware of one another, and sx varables f each person has feelngs for the other two. The varables need not be romatc love; the thrd person could be the chld of a couple or perhaps an n-law. Wth sx varables, there are 6 = 64 equlbra, only one of whch represents a unversally happy arrangement and 4 6 = 4096 dfferent combnatons of styles, assumng each person can adopt a dfferent nteracton style toward each of the others. Not surprsngly, the prognoss for coexstng postve feelngs s very low unless the ndvduals exhbt strong

8 8 Predator-Prey Dynamcs for Rabbts, Trees, and Romance adaptablty of ther styles. Perhaps humans adapt to such stutons much the way plants and anmals do, by lmtng ther nteractons, evolvng to fll a stable nche, drawng sustnence from others, and mantanng spatal separaton. Wth three or more varables, there s the possblty of chaotc solutons. However, a search for such solutons n the system of Eq. () generalzed to sx varables faled to reveal any such solutons, although other smlar models do exhbt chaos (Sprott 004). Acknowledgments I would lke to thank Warren Porter and Heke Lschke whose research nspred ths work and George Rowlands and Janne Bollger whose ongong collaboraton facltated ts completon. References Bak, P., 996, How Nature Works: The Scence of Self-organzed Crtcalty, Corperncus (New York). Chesson, P., 000, Mechansms of Mantenance of Speces Dversty. Annu. Rev. Ecol. Syst., 3, 343. Gause, G.F., 97, The Struggle for Exstence, Dover (New York). Lorenz, E.N., 963, Determnstc Nonperodc Flow. J. Atmos. Sc., 0, 30. May, R.M., 98, Theoretcal Ecology: Prncples and Applcatons, second ed., Blackwell Scentfc (Oxford). Murray, J. D., 993, Mathematcal Bology, Sprnger (Berln). Sprott, J.C., 004, Dynamcal Models of Love, Nonlnear Dynamcs, Psychology, and Lfe Scences 8, 303. Sprott, J.C., Bollger, J., and Mladenoff, D.J., 00, Self-organzed Crtcalty n Forestlandscape Evoluton. Phys. Lett. A, 97, 67. Sprott, J.C., Wldenberg, J.C., and Azz, Y., 005, A Smple Spatotemporal Chaotc Lotka- Volterra Model, Chaos, Soltons and Fractals 6, 035. Strogatz, S.H., 988, Love Affars and Dfferental Equatons. Mathematcs Magazne, 6, 35.