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The dynamics of arthropod predator-prey systems
BOOK REVIEW
Michall P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Monographs in Population Biology 13, Princeton U. P. 1978, 237 pp., Author index, index to genera cited, subject index, cloth $16.00, paper $6.95.
The mathematical study of species interactions has been devoted almost exclusively to the analysis of competition or predation. But these two kinds of relations have been studied in very different ways. Competition theory has been dominated by the coexistence problem: What determines the number of competing species that can coexist in a given environment? How similar can they be? What determines species diversity? Although most competitors are coconsumers of the same resources, and are therefore copredators in the broad sense, their common resources are usually abstracted from the problem, so that species linked through one or more prey species are visualized as being linked directly by a competitive linkage. Further, the problem has usually been translated into the static one: how many competitors can coexist in a stable equilibrium? Qualitative answers have been given to these questions: the number of species cannot exceed the number of discrete resources; on a continuum of resources, competing species cannot be too similar; “too similar” must be understood in terms of their niche breadths; unevenness of resource abundance increases the necessary ecological distance between species and reduces the number of species; the local stability of the equilibrium depends on the intensities of species interactions (competition coefficient). Although these coefficients vary with species abundances and the environment, they can be computed (in principle-many of the details are in dispute). But coexistence theory cannot be developed much further within a horizontal layer of the trophic structure. Nonequilibrium dynamics must deal with the dynamics of resource. utilization-predation-and even the static theory must look at various kinds of vertical linkages in the trophic hierarchy. Predation theory was from the start a nonequilibrium theory. The LotkaVolterra equations introduced conservative oscillations which depend on initial conditions, while further modifications of the model lead to equilibrium, limit cycle, and chaotic behaviors. The Lotka-Volterra system proved unsatisfactory both because its results were too sensitive to the assumptions of no self-damping and because the behavior of real predator-prey pairs did not seem to fit. And even when the Lotka-Volterra system is modified to allow for nonlinear behaviors, a pair of MATHEMATICAL
BIOSCIENCES
OElsevier North Holland, Inc., 1979.
46:303-306
303
(1979) 0025-5564/79/080303
+ 4$02.25
304
BOOK REVIEW
first-order autonomous differential equations simply does not allow the richness of behavior observed in real predator prey systems. There are a number of ways out of this dilemna: (1) The system could be expanded to include a larger number of variables. The additional variables may be other species, life-cycle stages, physiological states, or other components of ecosystems. Even the transition from two to three variables a new richness of possible behavior. (2) Some of the constant parameters of the system may be allowed to vary (e.g., seasonal development rates). (3) The differential-equation approach may be abandoned in favor of discrete-generation difference-equation models. Hassell does not discuss explicitly the reasons why he and the other investigators of the school which developed the approach presented in the monograph made the choice of the third alternative. However, several factors may be oeperating: First, there is the belief that the simplest systems which show interesting behavior provide a manageable way of learning about more complex communities. Second, there is no obvious way of representing environmental patterns. But if environmental variation is once allowed, the possible outcomes are almost unlimited, and it seems to be cheating (or violating some esthetic of simplicity) to derive oscillatory variables from oscillatory parameters. Third, the study of simple nonlinear difference equations developed together with the investigation of chaotic motion (nonperodic, nonequilibrated, bounded trajectories which look as if they were random processes), and the excitment of discovering such unexpected and interesting dynamics from simple models would be sufficient to attract further study. Finally, the main strategy in biological control is still the finding and encouragement of single, highly specific parasitoids which may in fact behave as the simple 2-variable autonomous difference equations. A further choice is made in the interest of simplicity: instead of dealing with predation, Hassell studies the parasitoid prey system, where search is limited to a single stage of the life cycle, there are no mixed diets, and the number of parasites produced is the number of prey parasitized. The result of these choices is a detailed study of the model N t+~=~,.f(NJ’t)~
P,+, = c&t1 -f(Nt,P,)l, in which N, and P, are the host and parasite
population,
respectively,
at
305
BOOK REVIEW
time r. The general function
f is usually
taken to be
so that the investigation sacrifices generality for precision while achieving moderate realism. Given these choices, Hassell develops the models from simple Nicholson-Bailey and functional-response situations through successive complications of nonrandom search, refugia, mutual interference. In each case, a standard procedure is followed. The test of stability for difference equations (eigenvalues less than 1 instead of less than zero) is applied. The answer comes out as an inequality among parameters. Thus we find regions of parameter space within which different kinds of dynamics would occur. Thus we learn that aggregation increases stability; that at high levels of the prey rate increase, stability breaks down; that increased spatial subdivision enhances stability. In addition, more complicated parameter relations are developed which are not easily described verbally but which can easily be read from the graphs of parameter space and which contribute toward developing our intuition about the properties of nonlinear difference equations. The standardization of procedure makes it easy to follow the development from case to case, but is also limiting. A number of questions which would help understand the dynamics of the systems, such as the amplitude and frequency of oscillation, are ignored in favor of the stability-instability paradigm. For instance, the very general form of the equations given above already leads to conclusions such as that the ratio of average prey population to average number of predators is X/c@- 1) independent of the form of f (N,P). Therefore, the smaller the reproductive rate of the prey (A), the smaller the proportion of the population parasitized. It can be further demonstrated that if f (N,P) is a monotone decreasing function of P (such as e-a), then parasitoid and prey are positively correlated over time. The purpose of these observations is not to complain that things were left out of the book, but to suggest that the emphasis on conditions for stability is an inheritance from past agendas which is now limiting. Perhaps the least satisfactory part of the book is the final chapter, “A theoretical basis for biological control,” in which the pest species are removed from the complex communities in which they live to be treated as part of isolated 2- or 3-species systems. The theoretical issue “Are pairwise interactions a sufficient basis for understanding community dynamics?” is thus placed on the agenda of ecology.
306
BOOK REVIEW
But more important than disagreements about research strategy is the historical importance of the work described by Hassell: agricultural entomology is now less dependent on borrowed technique and is creating its own mathematical framework and intuition; it is taking place in the context of collaborative efforts of model building and experiment, and its results are already powerful enough to undermine the narrow empiricism which dominates economic entomology and to justify theoretical research. RICHARD
LEVINS
School of Public Health Harvard University Boston, Massachusetts
Michall P. Hassell, The Dynamics of Arthropod Predator-Prey Systems, Monographs in Population Biology 13, Princeton U. P. 1978, 237 pp., Author index, index to genera cited, subject index, cloth $16.00, paper $6.95.
The mathematical study of species interactions has been devoted almost exclusively to the analysis of competition or predation. But these two kinds of relations have been studied in very different ways. Competition theory has been dominated by the coexistence problem: What determines the number of competing species that can coexist in a given environment? How similar can they be? What determines species diversity? Although most competitors are coconsumers of the same resources, and are therefore copredators in the broad sense, their common resources are usually abstracted from the problem, so that species linked through one or more prey species are visualized as being linked directly by a competitive linkage. Further, the problem has usually been translated into the static one: how many competitors can coexist in a stable equilibrium? Qualitative answers have been given to these questions: the number of species cannot exceed the number of discrete resources; on a continuum of resources, competing species cannot be too similar; “too similar” must be understood in terms of their niche breadths; unevenness of resource abundance increases the necessary ecological distance between species and reduces the number of species; the local stability of the equilibrium depends on the intensities of species interactions (competition coefficient). Although these coefficients vary with species abundances and the environment, they can be computed (in principle-many of the details are in dispute). But coexistence theory cannot be developed much further within a horizontal layer of the trophic structure. Nonequilibrium dynamics must deal with the dynamics of resource. utilization-predation-and even the static theory must look at various kinds of vertical linkages in the trophic hierarchy. Predation theory was from the start a nonequilibrium theory. The LotkaVolterra equations introduced conservative oscillations which depend on initial conditions, while further modifications of the model lead to equilibrium, limit cycle, and chaotic behaviors. The Lotka-Volterra system proved unsatisfactory both because its results were too sensitive to the assumptions of no self-damping and because the behavior of real predator-prey pairs did not seem to fit. And even when the Lotka-Volterra system is modified to allow for nonlinear behaviors, a pair of MATHEMATICAL
BIOSCIENCES
OElsevier North Holland, Inc., 1979.
46:303-306
303
(1979) 0025-5564/79/080303
+ 4$02.25
304
BOOK REVIEW
first-order autonomous differential equations simply does not allow the richness of behavior observed in real predator prey systems. There are a number of ways out of this dilemna: (1) The system could be expanded to include a larger number of variables. The additional variables may be other species, life-cycle stages, physiological states, or other components of ecosystems. Even the transition from two to three variables a new richness of possible behavior. (2) Some of the constant parameters of the system may be allowed to vary (e.g., seasonal development rates). (3) The differential-equation approach may be abandoned in favor of discrete-generation difference-equation models. Hassell does not discuss explicitly the reasons why he and the other investigators of the school which developed the approach presented in the monograph made the choice of the third alternative. However, several factors may be oeperating: First, there is the belief that the simplest systems which show interesting behavior provide a manageable way of learning about more complex communities. Second, there is no obvious way of representing environmental patterns. But if environmental variation is once allowed, the possible outcomes are almost unlimited, and it seems to be cheating (or violating some esthetic of simplicity) to derive oscillatory variables from oscillatory parameters. Third, the study of simple nonlinear difference equations developed together with the investigation of chaotic motion (nonperodic, nonequilibrated, bounded trajectories which look as if they were random processes), and the excitment of discovering such unexpected and interesting dynamics from simple models would be sufficient to attract further study. Finally, the main strategy in biological control is still the finding and encouragement of single, highly specific parasitoids which may in fact behave as the simple 2-variable autonomous difference equations. A further choice is made in the interest of simplicity: instead of dealing with predation, Hassell studies the parasitoid prey system, where search is limited to a single stage of the life cycle, there are no mixed diets, and the number of parasites produced is the number of prey parasitized. The result of these choices is a detailed study of the model N t+~=~,.f(NJ’t)~
P,+, = c&t1 -f(Nt,P,)l, in which N, and P, are the host and parasite
population,
respectively,
at
305
BOOK REVIEW
time r. The general function
f is usually
taken to be
so that the investigation sacrifices generality for precision while achieving moderate realism. Given these choices, Hassell develops the models from simple Nicholson-Bailey and functional-response situations through successive complications of nonrandom search, refugia, mutual interference. In each case, a standard procedure is followed. The test of stability for difference equations (eigenvalues less than 1 instead of less than zero) is applied. The answer comes out as an inequality among parameters. Thus we find regions of parameter space within which different kinds of dynamics would occur. Thus we learn that aggregation increases stability; that at high levels of the prey rate increase, stability breaks down; that increased spatial subdivision enhances stability. In addition, more complicated parameter relations are developed which are not easily described verbally but which can easily be read from the graphs of parameter space and which contribute toward developing our intuition about the properties of nonlinear difference equations. The standardization of procedure makes it easy to follow the development from case to case, but is also limiting. A number of questions which would help understand the dynamics of the systems, such as the amplitude and frequency of oscillation, are ignored in favor of the stability-instability paradigm. For instance, the very general form of the equations given above already leads to conclusions such as that the ratio of average prey population to average number of predators is X/c@- 1) independent of the form of f (N,P). Therefore, the smaller the reproductive rate of the prey (A), the smaller the proportion of the population parasitized. It can be further demonstrated that if f (N,P) is a monotone decreasing function of P (such as e-a), then parasitoid and prey are positively correlated over time. The purpose of these observations is not to complain that things were left out of the book, but to suggest that the emphasis on conditions for stability is an inheritance from past agendas which is now limiting. Perhaps the least satisfactory part of the book is the final chapter, “A theoretical basis for biological control,” in which the pest species are removed from the complex communities in which they live to be treated as part of isolated 2- or 3-species systems. The theoretical issue “Are pairwise interactions a sufficient basis for understanding community dynamics?” is thus placed on the agenda of ecology.
306
BOOK REVIEW
But more important than disagreements about research strategy is the historical importance of the work described by Hassell: agricultural entomology is now less dependent on borrowed technique and is creating its own mathematical framework and intuition; it is taking place in the context of collaborative efforts of model building and experiment, and its results are already powerful enough to undermine the narrow empiricism which dominates economic entomology and to justify theoretical research. RICHARD
LEVINS
School of Public Health Harvard University Boston, Massachusetts