The biological control paradox

TREE vol. 6, no. 7, January

1991

trlologicalcontrol Paradox Luck’ raises a paradox for the biological control of pests by natural enemies. On one hand, conventional mathematical models predict that you cannot have a prey equilibrium that is both low and highly stable. On the other hand, there are numerous examples of successful biological control where the prey are maintained at densities less than 2% of their carrying capacities. Luck also mentions that controlled pests do not remain in a strict equilibrium; they often appear to fluctuate because of local extinctions. The paradox arises from the structure of conventional predator-prey models which give rise to a humped (parabolic) prey isocline and a vertical rectilinear predator isocline2. Because of this latter property, the prey equilibrium is independent of prey parameters. Its density and stability are entirely dependent on predator characteristics, with efficient predators creating low, unstable prey equilibria and large oscillations that periodically bring the prey population to values close to its carrying capacity3 (Fig. la). A number of arguments suggest that the predator isocline should often be slanted rather than vertical. If the prey population is maintained at a higher density, say by stocking, then the predator population should attain a higher equilibrium4r5. Also, prey and predator densities should both be correlated with prey productivity6. Slanting isoclines are obtained if the logistic model is extended to the consumer leve15,‘**, if the functional response depends on the prey/predator ratio6sg or, more generally, if the per capita rate of increase of predators declines with their own density1&12. With this isocline structure, the prey equilibrium can be reduced and its stability increased by changing prey parameters, e.g. by reducing its intrinsic rate of increase or its vulnerability to predation (Fig. 1 b).

More importantly, if prey parameters cannot be changed, the introduction of efficient predators can control the prey by producing the

) (b)

local pattern shown in Fig. Ic. All trajectories enter sector S3 where the populations are driven to extinction. However, when populations become small, environmental noise or pest immigration can easily bring the trajectories back into sector Sl where both populations grow again. Thus, this graphical model makes three qualitative predictions that are compatible with the patterns described by Luck for populations under biological control: (1) both populations are maintained at low densities; (2) no stable equilibrium exists; (3) local populations experience repeated quasi-extinctions. The family of models producing slanted isoclines is not derived from detailed mechanistic (‘microscopic’) assumptions about the behavior of prey and predator individuals. Thus, they cannot replace the line of research reviewed by Luck, which has clarified the role of various patterns of behavior (aggregation, etc.). What they can do is provide simple phenomenological (‘macroscopic’) characterizations of biological control. They also solve the biological control paradox and, interestingly, Rosenzweig’s paradox of enrichment as wel13f6. Roger Arditi

Prey abundance

K

Fig. 1. Phase plane analysis represents the direction of variation of prey and predator abundances in time. The prey isocline is the line where the rate of variation of prey (with respect to predators) is zero. K is the prey carrying capacity. (a) Conventional models with vertical predator isoclines; (b,c) models with slanted predator isoclines. Dashed lines show the effect of increasing prey vulnerability or lowering its rate of increase: in (a) the prey equilibrium remains the same, while in (b) it is reduced. In (a) low equilibria are unstable and give rise to limit cycles; in (b) low stable equilibria are possible. Case (c) illustrates non-equilibrium biological control at the lower level: if the predator is efficient enough, the isoclines intersect at the origin, the populations remain at low densities and exhibit growth phases followed by quasiextinctions.

Institute of Zoology and Animal Ecology, University of Lausanne, CH-1015 Lausanne, Switzerland

Alan A. Berryman Dept of Entomology, Washington State University, Pullman, WA 99X4-6432, USA

R.F. (1990) Trends Ecol. Evol. 5, 196-199 2 Rosenzweig, M.L. and MacArthur, R.H. (I 963) Am. Nat. 97.209-223 3 Rosenzweig, M.L. (1971) Science 171 385-387 A.A. (1981) Population 4 Berryman, Systems: A Genera/ Introduction, 1 Luck,

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5 Berryman, A.A. and Stenseth, N.C. (1984) Behav. SC;. 29. 127-137 6 Arditi, R. and Ginzburg, L.R. (1989) J. Theor. Biol. 139, 31 l-326 7 Leslie, P.H. (1948) Biometrika 35, 213-245 8 Berryman, A.A. (1990) Population Analysis System: POPSYS Series 2, TwoSpecies Analysis, Ecological Systems Analysis 9 Arditi, R. and AkFakaya, H.R. (1990) Oecoloaia 83,358-361 10 DeAngelis, D.L., Goldstein, R.A. and O’Neill, R.V. (1975) Ecology 56,881-892 11 Getz, W.M. (1984) J. Theor. Biol. 108,

623-643 12 Gutierrez, and Summers, 116,923-963

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A.P., Baumgaertner, J.U. C.G. (1984) Can. Entomoi.