The Volterra Principle Generalized Erschienen in: Philosophy of Science ; 84 (2017), 4. - S. 737-760 https://dx.doi.org/10.1086/693874 The Volterra Principle Generalized Tim Räz*y Michael Weisberg and Kenneth Reisman argue that the Volterra Principle can be derived from multiple predator-prey models and that, therefore, the Volterra Principle is a prime example for robustness analysis. In the current article, I give new results regarding the Volterra Principle, extending Weisberg’s and Reisman’s work, and I discuss the conse- quences of these results for robustness analysis. I argue that we do not end up with mul- tiple, independent models but rather with one general model. I identify the kind of situ- ation in which this generalization approach may occur, and I analyze the generalized Volterra Principle from an explanatory perspective. 1. Introduction. The Lotka-Volterra predator-prey model is historically important because it initiated mathematical modeling in biology. It is also philosophically important because it is a much-discussed case study in the debatesonscientificmodeling,explanation,idealization,androbustnessanal- ysis. One of the most important results of the Lotka-Volterra predator-prey model is the Volterra Principle. This principle, first formulated as the Third Law by Vito Volterra, is supposed to explain an empirical regularity about fishery statistics. However, the Lotka-Volterra predator-prey model is highly idealized, to the point that a realistic interpretation of the model is question- able. Does this mean that the Volterra Principle is questionable as well? Weisberg (2006b), and Weisberg and Reisman (2008) argue that this is not the case. They propose that the Volterra Principle can also be established from *To contact the author, please write to: FB Philosophie, Universität Konstanz, 78457 Konstanz, Germany; e-mail: tim.raez@gmail.com. yI thank Claus Beisbart, Matthias Egg, Nick de Hoog, Mark Kot, Andrea Loettgers, Thomas Müller, Antje Rumberg, Tilman Sauer, Verena Wagner for comments on various drafts of the article and Dan Ward for proofreading. This work was supported by the Templeton World Charity Foundation through grant TWCF0078/AB46. Received September 2016; revised January 2017. Philosophy of Science, 84 (October 2017) pp. 737–760. 0031-8248/2017/8404-0007$10.00 Copyright 2017 by the Philosophy of Science Association. All rights reserved. This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License (CC BY-NC 4.0), which permits non-commercial reuse of the work with attribution. For commercial use, contact journalpermissions@press.uchicago.edu. 737 Konstanzer Online-Publikations-System (KOPS) URL: http://nbn-resolving.de/urn:nbn:de:bsz:352-1-e05466f82145418a0 738 TIM RÄZ other predator-prey models. According to Weisberg, the Volterra Principle is thus a prime example for the practice of robustness analysis. In the current article, I present new results concerning the Volterra Prin- ciple, extending and refining Weisberg’s and Reisman’s work, and I discuss the consequences of these new results for robustness analysis. The results are in keeping with the goals of robustness analysis proposed by Weisberg. However, they deviate from the process of robustness analysis that Weisberg proposes. In particular, we do not end up with multiple independent models but rather with one general model. I identify the situations in which this gen- eralization approach may apply. I analyze the generalized Volterra Principle from an explanatory perspective, and I show that my results exhibit several virtues of mathematical explanations. Finally, I discuss cases in which the generalization approach may not apply, and I propose that these are in fact cases of mathematical coincidences. 2. The Lotka-Volterra Predator-Prey Model. The Lotka-Volterra predator- prey model was originally proposed by Alfred Lotka and Vito Volterra.1 Formally, the model is a set of two coupled ordinary differential equations. These equations are intended to capture the interactions between a pop- ulation of prey animals and a population of predators, which feed on the prey.2 The model can be written as follows: _x 5 rx 2 axy: (1) _y 5 bxy 2 my: (2) Here, x is shorthand for x(t), the size of the prey population at time t; y is shorthand for y(t), the size of the predator population at time t; _x and _y are the respective time derivatives. According to equation (1), the prey population x grows at a constant positive rate r in the absence of predators; in the presence of predators, the number of prey decreases proportionally to the number of predators, at a constant positive rate a. According to equation (2), the pred- ator population y decreases at a constant positive rate m in the absence of prey; in the presence of prey, the predator population grows proportionally to the number of prey available, at a constant positive rate b. The system is coupled because both equations depend on both x and y. Dynamically, this system exhibits undampened oscillations: if undisturbed, it will oscillate in- definitely on the same circular orbit in phase space (see fig. 1). 1. See Knuuttila and Loettgers (2016) for a discussion of the contrast between Lotka’s and Volterra’s approach to scientific modeling in general and to their eponymous model in particular. 2. Kot (2001) gives a detailed and accessible introduction to the mathematical aspects of the Lotka-Volterra model and predator-prey models in general. VOLTERRA PRINCIPLE GENERALIZED 739 Volterra’s motivation for formulating the predator-prey model was an em- pirical phenomenon in need of explanation.3 The explanandum concerns fishery statistics. During World War I, when fishing in the Adriatic was di- minished, fishery statistics showed that the ratio between predators and prey fish shifted in favor of predators. After the war, when fishing returned to its prewar intensity, the ratio shifted back, and favored the prey. This prompted the question as to why this shift in proportions occurred. The explanation is provided by Volterra’s Third Law, now called the Volterra Principle. The Vol- terra Principle states that, if we continuously remove a constant proportion of both the predator and prey populations, then the average number of pred- ators will decrease relative to the average number of prey (see fig. 2). Vol- terra showed that this shift of averages can be derived from the predator-prey model.Hethusprovidedahow-possiblyexplanationofthestatisticalphenom- enon: the regularity holds in a very simple model of population dynamics.4 Figure 1. Periodic orbits in phase space: N1 denotes the prey population; N2 denotes the predator population. Orbits X, W, L, F are examples of the exhaustive and disjunct orbits corresponding to systems with different initial conditions; Q is the equilibrium state of all systems. If a system is not disturbed, it evolves on its orbit, running coun- terclockwise with a constant period, and it stays on its orbit; this is Volterra’s First Law. Illustration from Volterra (1928, 14): by permission of Oxford University Press. 3. See Volterra (1928) for Volterra’s own account of the story. 4. See, e.g., Forber (2010) and Scholl and Räz (2013) on the notion of how-possibly ex- planations. 740 TIM RÄZ It has, however, been disputed that the Volterra Principle provides an ac- tual explanation. Many mathematical population biologists today do not be- lieve that the Lotka-Volterra model is a realistic representation of predator- prey interactions. The main mathematical problem with the model is that it is not structurally stable: if perturbations terms are added to the model equa- tions, the model’s qualitative behavior changes.5 Does this mean that the Vol- terra Principle is also doomed to fail? This question has been brought to phi- losophers’ attention in two papers (Weisberg 2006b; Weisberg and Reisman 2008). Weisberg and Reisman argue that the Volterra Principle does not de- Figure 2. Population density of the predator-prey system with orbit L at the equi- librium point Q on average; this is Volterra’s Second Law. If equations (1) and (2) of the system at point P are changed, such that the predator death rate m increases and the prey birth rate r decreases, the system evolves on the new orbit L0 with the equi- librium point Q0. On average, system L0 has a higher prey density (K1 to K01) and a lower predator density (K2 to K 0 2). This is Volterra’s Third Law. Before and after World War I, the system was on orbit L0; during World War I, with greatly reduced fishing, the system was on orbit L. Illustration from Volterra (1928, 19): by permis- sion of Oxford University Press. 5. See Brauer and Castillo-Chávez (2001, 180) for more on the notion of structural sta- bility. VOLTERRA PRINCIPLE GENERALIZED 741 pend on the specifics of the original predator-prey model. Rather, the Volterra Principle can be derived from a variety of models, which are more realistic than the original predator-prey model. In short, Weisberg and Reisman claim that the Volterra Principle is a “robust theorem.” This makes the Volterra Prin- ciple a prime example for the practice of robustness analysis. In the next sec- tion, I give a brief introduction to robustness analysis, as well as to Weisberg and Reisman’s account of the Volterra Principle. 3. Robustness Analysis 3.1. Early Contributions. Robustness analysis was first discussed as a scientific practice in a seminal paper by Levins (1966).6 Levins observes that population biology makes use of highly idealized models; that is, the models are based on assumptions that are, strictly speaking, false with respect to their target systems. How can we make sure, nonetheless, that results we de- rive from such models are still true with respect to their target systems and not mere artifacts of the idealizations of the models? Levins recommends ro- bustness analysis as a remedy. The idea is to derive results from “several al- ternative models each with different simplifications but with a common bi- ological assumption” (423). If this procedure is carried out successfully, the results are called “robust theorems.” The slogan of robustness analysis is that “our truth is the intersection of independent lies” (423). An early adopter of robustness analysis is Wimsatt (2012b; see also Wim- satt 2007). Wimsatt advocates a wide conception of robustness analysis, also dubbed “multiple determination,” whereby the practice is not restricted to the derivation of phenomena from different models but rather encompasses, say, the detection of phenomena using different senses, or different experi- mental methods. Wimsatt emphasizes that it is important to determine the limitations of robustness results, given that robustness analysis involves, in part, understanding the scope and limitations of the robustness of a phe- nomenon. An important critique of Levins’s proposal has been formulated by Or- zack and Sober (1993). Orzack and Sober point out that robustness analysis cannot play a confirmatory role. If all the models used in robustness analysis are idealized, then we do not know whether results derived from these mod- els are true with respect to the target system. If we want to be certain, we would have to make sure that at least one of the models faithfully represents the target system. However, this is usually not the case. In short, independent lies are still lies. In reaction to Orzack’s and Sober’s objection, Levins (1993) argues that he did not intend robustness analysis to provide nonempirical 6. See Weisberg (2006a) for an appraisal of Levins’s paper. 742 TIM RÄZ confirmation; rather, robustness analysis is merely supposed to help in deter- mining the scope and limits of a result. 3.2. Weisberg and Reisman. The most influential recent account of ro- bustness analysis has been proposed by Weisberg (2006b).7 Weisberg aims to reconcile robustness analysis with the objections raised by Orzack and So- ber. He proposes that robustness analysis does not confer full-blown confir- mation but rather “low-level confirmation,” that is, “confirmation of the fact that certain mathematical structures can adequately represent properties of target phenomena” (740). Weisberg characterizes robustness analysis as a four-stepprocess.Inthefirststep,oneexamineswhetherseveraldistinctmod- els share a property that is expected to be robust. In the second step, one tries to identify the “core structure” that is responsible for the robust property. The third step consists of an empirical interpretation of the mathematical struc- ture. The fourth step consists of “stability analysis,” with the purpose of find- ing out how sensitive the models are with respect to slight changes to their structure. Weisberg uses the Volterra Principle as a case study to support his anal- ysis. He argues that the Volterra Principle is robust, given that it can be ob- tained in different predator-prey models.8 Specifically, Weisberg claims, we can modify the structure of the original predator-prey model by adding den- sity dependence and predator satiation and still obtain the Volterra Principle; see the appendix for a brief discussion of these models. Weisberg claims that the Volterra Principle can be shown to be valid for both of these models. Es- tablishing that these two models satisfy the Volterra Principle corresponds to the first step of robustness analysis. Weisberg then argues that these models have a common structure: the growth rate of predators is mostly controlled by the growth rate of prey, and the growth rate of prey is mostly controlled by the death rate of predators. According to Weisberg, this common structure is responsible for the phenomenon that predators and prey are affected differ- ently bycontinuousdestructionofboth species.Theidentification ofthecom- mon structure concludes step 2 of robustness analysis. Weisberg and Reisman (2008) push robustness analysis further, and they refine the results regarding the Volterra Principle. First, they distinguish three kinds of robustness. Parameter robustness investigates robustness with re- spect to a variation of parameters in a model. For example, the Volterra Prin- ciple is robust under variations of the parameters a, b, r, and m in models (1) 7. The interest in Levins’s paper, and robustness analysis, was renewed by Weisberg and Odenbaugh (2006). These papers initialized the “second wave” of robustness analysis. See Wimsatt (2012a) for a useful account of the development of robustness analysis. 8. Weisberg’s ideas are based on suggestive remarks in Roughgarden (1979), a textbook in population ecology. VOLTERRA PRINCIPLE GENERALIZED 743 and (2). Structural robustness encompasses the more general robustness across models with different (causal) structure (e.g., models with or without density dependence).9 Representational robustness encompasses models that are for- mulated in different frameworks (e.g., using continuous vs. discrete vari- ables). Weisberg and Reisman then argue that the Volterra Principle is robust according to all three kinds of robustness. They provide a formulation of the Volterra Principle that modifies the “hypothesis” from Weisberg (2006b). The new formulation is based on the notions of negative coupling and bio- cide. A predator-prey system is negatively coupled “just in case increasing the abundance of predators decreases the abundance of prey and increasing the abundance of prey increases the abundance of predators” (Weisberg and Reisman 2008, 114). In a general biocide, the death rate of predators is in- creased and the birth rate of the prey is decreased.10 The new formulation of the Volterra Principle reads as follows: 9. St notio 10. I made (VP) Ceteris paribus, if a two-species, predator-prey system is negatively coupled, then a general biocide will increase the abundance of the prey and decrease the abundance of predators. (114) Weisberg and Reisman claim that this principle holds in general. How- ever, they do not provide a conclusive answer as to the scope and limits of VP. Weisberg and Reisman analyze further models with respect to the Vol- terra Principle. In Weisberg (2006b), robustness analysis was restricted to parameter and structural robustness—the models are all formulated in the framework of coupled ordinary differential equations—while Weisberg and Reisman (2008) also includes an analysis of individual-based models (IBMs). These are finite models with discrete space and time, which can be taken to represent individual animals. In the IBMs considered by Weisberg and Reis- man, the discrete transition rules are not deterministic but rather stochastic. The behavior of these models is investigated via simulations. Weisberg and Reisman argue that the Volterra Principle can be recovered in this framework, and they therefore conclude that the Volterra Principle is representationally ro- bust. 3.3. After Weisberg and Reisman. The work by Weisberg and Reisman has renewed philosophers’ interest in robustness analysis. Weisberg’s ac- count of robustness analysis and the notion of low-level confirmation have ructural robustness is not to be confused with structural stability, a mathematical n. n the case of systems (1) and (2), r is decreased and m is increased. The notion is precise in definition 1, see sec. 4.1 below. 744 TIM RÄZ received considerable attention. Calcott points out that a robust theorem is only as good as the set of models from which it has been derived and that this is not sufficient for confirmation. However, robustness can neverthe- less be helpful by “narrowing the possibility space of candidate solutions” (Calcott 2011, 287). Houkes and Vaesen (2012) provide an in-depth analysis of Weisberg’s work. The main lesson of their paper is that the evaluative role of robustness analysis concerning a set of models is only as good as the cred- ibility of the model family from which the set of models is chosen. The idea of robustness analysis has also been applied to pure mathematics: Krömer (2012) discusses a potential role for robustness in the foundations of math- ematics, and Corfield (2010) focuses on mathematical structures exhibiting a surprising “confluence” of structural properties. The idea of robustness anal- ysis has also been applied to cases from biology in Knuuttila and Loettgers (2011) and to economics in Kuorikoski, Lehtinen, and Marchionni (2010). Robustness analysis is also relevant to the issue of scientific and mathe- matical explanation. The idea that robustness analysis can enhance our un- derstanding of a phenomenon was proposed early in the literature (e.g., by Wimsatt 2012b). However, the relevance of robustness analysis for explana- tionhasnotbeen systematicallyexplored.A notable exceptionisa recentcon- tribution by Schupbach (2016). Schupbach proposes that robustness analysis can be interpreted as an explanatory enterprise. The idea is that if we establish the robustness of a property, then we can rule out competing explanations of that property. Schupbach illustrates the idea with the Volterra Principle. Ini- tially, we should not be confident that the Volterra Principle is generally valid because we have only derived it from the original predator-prey model, and the result could be due to the idealizations of this simple model. However, if we can establish the Volterra Principle for models with different idealizations, then, according to Schupbach, we can rule out certain competing explana- tions as to why the Volterra Principle holds, namely, explanations attributing this fact to idealizations that are not present in the new models. 3.4. Robustness Analysis and Mathematics. In the current article, I re- strict my attention to one particular kind of robustness, namely, the case of robust theorems that are derived from multiple independent models. This is a special case of robustness, in comparison to the wider notion of robustness championed, for example, by Wimsatt.11 The focus of the discussion will be on the role of mathematics in robustness analysis. While it has been acknowl- edged in the philosophical discussion that mathematics has an important role to play in robustness analysis, a systematic exploration of this role is so far missing from the discussion. I strive to fill this lacuna here. I will argue that 11. See Calcott (2011) for a similar useful distinction between several kinds of robust- ness. VOLTERRA PRINCIPLE GENERALIZED 745 mathematics can help us enhance our confidence in a robust result by show- ing that it does not depend on particular idealizations. However, this is not achieved by deriving the result from multiple independent models but rather by deriving the result from one general model. I will not argue that mathemat- ics, or robustness analysis, can play a confirmatory role. My discussion of ro- bustness analysis is based on scientific practice: in the next section, I present new results concerning the Volterra Principle. 4. The Volterra Principle Generalized 4.1. The Volterra Principle in a Gause-Type Model. Here, I generalize the results in Weisberg (2006b) regarding the Volterra Principle. More spe- cifically, I prove that the Volterra Principle holds for a more general type of model, which was first proposed by Gause (1934). I use the following mod- ification of a Gause-type model:12 _x 5 xrf xð Þ 2 p xð Þy, (3) _y 5 p xð Þy 2 my, (4) where x is prey abundance, y is predator abundance, r is the constant part of the prey growth rate, and m is predator mortality. The prey growth rate func- tion f(x), which was missing in the original predator-prey model, is assumed to be differentiable for x ≥ 0; additionally, it is assumed that df =dx ≕ fx ≤ 0 and f (0) 5 a > 0. The environment can have a carrying capacity, that is, a constant K > 0 with f (K) 5 0. The functional response p(x), which was as- sumed to be the identity function in the original predator-prey model, is as- sumed to be differentiable for x ≥ 0; additionally, it is assumed that dp= dx ≕ px > 0 and p(0) 5 0.13 The Volterra Principle states how the equilibrium of a dynamical system is affected by a change in the parameters of the system (see fig. 2). In order for this to be biologically meaningful, systems (3) and (4) need to have an equilibrium inside the first quadrant (i.e., an equilibrium with positive pop- ulation variables). For the existence of such an equilibrium, we have to as- sume that there is an x* such that p(x*) 5 m. For the formulation of the Volterra Principle, we additionally assume the existence of an x0 with p(x0) 5 m 1 d for a given d > 0. Values x* and x0 are unique because p(x) is increas- 12. See Freedman (1980, chap. 4). In comparison to system (4.2) (Freedman 1980, 66), I use rf(x) instead of g(x) in eq. (3) and p(x) instead of cp(x) in eq. (4). 13. The numerical response, p(x) in (4), is set equal to the functional response, p(x) in (3). See Freedman (1980, chap. 4) for a further discussion of the significance of these assumptions. 746 TIM RÄZ ing. If the system has a carrying capacity K, we have to assume that x*, x0 < K for the existence of an equilibrium. The coordinates of the equilibrium in the first quadrant can be determined as follows: we set equations (3) and (4) equal to 0 and solve the resulting equations. Using the assumption that there is a unique x* with p(x*) 5 m, we can deduce that the unique equilibrium has the following coordinates: x*, x*rf x*ð Þ p x*ð Þ � � : (5) The change in parameters, which leads to a change in equilibrium, can be interpreted as an intervention. The following definition makes this precise: Definition 1. A Volterra Intervention in a predator-prey system is a de- crease of r by an amount ε, with 0 < ε < r, and an increase of m by an amount d > 0. The assumption that 0 < ε < r avoids a negative growth rate. And 0 < d yields a new unique equilibrium in the positive quadrant after the Volterra Intervention. If we apply a Volterra Intervention to equilibrium (5), and use p(x0) 5 m 1 d, we get a unique equilibrium with the following coordinates: x0, x0 r 2 εð Þf x0ð Þ p x0ð Þ � � : (6) We are interested in the relation between equilibrium (5) and equilibrium (6); more specifically, how a Volterra Intervention affects the relative abundance of the populations: Definition 2. A predator-prey system has the Volterra Property if a Vol- terra Intervention, applied to an equilibrium (x*, y*), yields an equilibrium (x0, y0) such that the equilibrium abundance of predators relative to prey (i.e., their ratio) decreases: y*=x* > y0=x0 If we examine figure 2, we see that the original Lotka-Volterra model has the Volterra Property: K2=K1 > K 0 2=K 0 1 is true because K2 > K 0 2 and K1 < K 0 1. We are now in the position to formulate the following theorem: Theorem 1. For Gause-type models (3) and (4), the Volterra Property holds true. Proof. We have to show that the Volterra Property holds true for sys- tems (3) and (4); that is, the ratio y*=x* from equilibrium (5) is bigger than the ratio y0=x0 from equilibrium (6): 14. S VOLTERRA PRINCIPLE GENERALIZED 747 rf x*ð Þ p x*ð Þ > r 2 εð Þf x0ð Þ p x0ð Þ : (7) This inequality can be rearranged to yield f x*ð Þ f x0ð Þ p x0ð Þ p x*ð Þ > 1 2 ε r : (8) Because p(x*) 5 m, p(x0) 5 m 1 d, and because p is strictly increasing, we know that x0 > x*; that is, x0 5 x* 1 k for some k > 0. Replacing x0 with x* 1 k, we see that (8) is equivalent to f x*ð Þ f x* 1 kð Þ p x* 1 kð Þ p x*ð Þ > 1 2 ε r : (9) The fact that f is a decreasing function implies that f (x*)=f (x* 1 k) ≥ 1, and the fact that p is a strictly increasing function implies that p(x* 1 k)= p(x*) > 1. This implies that the left-hand side of the inequality is bigger than 1. The right-hand side, however, is smaller than 1 for ε > 0. Therefore, the inequality holds true. QED 4.2. Stable Equilibria. Theorem 1 is a statement about how the equilib- rium of a system changes under a Volterra Intervention. However, this is not sufficient for the result to be biologically meaningful. The existence of an equilibrium does not guarantee that the average abundance of a system co- incides with the equilibrium. In the case of the original Lotka-Volterra model, this was guaranteed by Volterra’s Second Law (see fig. 2). However, other predator-prey models show a variety of qualitatively different behaviors. For example, the so-called Samuelson model has an equilibrium that is stable or unstable depending on the choice of parameters.14 For some choices, the system spirals toward a stable equilibrium in the first quadrant. For other choices, the system spirals away from an unstable equilibrium. In the latter case, it is not relevant whether the Volterra Principle holds true because the equilibrium point is not where the system is on average. Rather, the system will show more and more violent fluctuations, such that the equilibrium shift predicted by the Volterra Principle is irrelevant. The same problem occurs in models with limit cycles. For example, the model with predator satiation ex- amined in Weisberg (2006b, 736) satisfies the Volterra Principle according to theorem 1. However, the system does not converge toward an equilibrium be- cause the model exhibits limit cycles, and it is not clear how average abun- dance and equilibrium are related—a result analogous to Volterra’s Second Law would be necessary to establish such a relation. ee Freedman (1980, 37) for a discussion of the Samuelson model. 748 TIM RÄZ One way of making sure that the equilibrium is biologically relevant is to require that it is stable.15 Stability questions are standard in mathematical pop- ulation ecology, and theorem 1 can be combined with an available stability criterion. An eigenvalue analysis of generalized Gause systems yields a crite- rion for the equilibrium’s stability (see Freedman 1980, 72, condition [4.23]). This criterion implies that the relevant equilibrium of systems (3) and (4) is stable under the following condition:16 H x*ð Þ 5 x*fx x*ð Þ 1 f x*ð Þ 2 x*f x*ð Þpx x*ð Þ p x*ð Þ < 0: (10) This condition can be combined with theorem 1 as follows: if the equilibrium of systems (3) and (4), before and after the Volterra Intervention, is stable ac- cording to condition (10), then theorem 1 is certainly biologically meaningful. 4.3. The Limits of the Volterra Principle. An important advantage of theorem 1 is that condition (9) allows us to infer the limits of the Volterra Principle. In particular, condition (9) shows how the proof of the theorem depends on the assumptions about functions f and p, and it also shows that these assumptions are minimal, in that if they are weakened further, the the- orem does not hold. Here I will discuss two ways in which the Volterra Prin- ciple does not hold if the assumptions on f and p are weakened. 15. T equil struc asym 16. T Possibility 1: Assume that fx > px > 0 in a range x ∈ ½x*, x* 1 k�; that is, f is no longer decreasing but increasing faster than p. This implies that the left-hand side of (9) gets smaller than 1 and the inequality is violated for sufficiently small ε. Possibility 2: Assume that px < 0 (i.e., the functional response decreases). In this case, we have to be more careful because the theorem depends on px > 0 in several places. First, we found that px > 0 implies the unique- ness of the equilibrium. If px < 0, there could be two equilibria or none. Second, we used px > 0 to infer x 0 > x*. If px < 0, this relation is inverted; this leads to a modification of condition (9): f x*ð Þ f x* 2 kð Þ p x* 2 kð Þ p x*ð Þ > 1 2 ε r : (9 0 )(90) he relevant notion here is asymptotic stability, which means that systems close to ibrium will eventually converge to that equilibrium. This is not to be confused with tural stability mentioned above. See Freedman (1980) and Kot (2001) for more on ptotic stability. he derivative of f, p with respect to x, evaluated at x*, is fx(x*), px(x*). VOLTERRA PRINCIPLE GENERALIZED 749 If we assume that p does not change sign, we need fx > jpxj > 0 in the range x ∈ ½x* 2 k, x*� in order to violate inequality (90) (i.e., f has to increase faster than p is decreasing).17 Are the conditions under which the Volterra Principle is violated biolog- ically realistic? Consider, first, possibility 1: fx > 0 is not realistic globally because it implies unbounded growth of prey in the absence of predators. However, fx > 0 seems reasonable in a limited range, in particular when the prey population is small and growth is not restricted externally. Turning to possibility 2, the case of a decreasing functional response p has been dis- cussed in the literature (see, e.g., Kot 2001, 140). According to Kot, so- called type IV functional responses can occur because of prey interference or prey toxicity. Thus, both possibilities potentially occur in situations that are biologically realistic. It could be asked whether the assumptions of theorem 1, on the one hand, and the requirement of stable equilibria, on the other, are independent. Does the theorem exclude unstable equilibria, or can we find predator-prey models that violate the Volterra Principle and have stable equilibria at the same time? In order to answer this question, it is useful to reformulate the stability con- dition (10). If we assume that f(x*) is not 0, (10) is equivalent to fx x*ð Þ f x*ð Þ 2 px x*ð Þ p x*ð Þ < 2 1 x* : (11) The expression on the left-hand side contains statements about the rela- tive change in f and p. The question now is whether there are cases in which condition (11) holds true while condition (9) is violated. Consider possibility 1 of violating (9); that is, fx > px > 0. This is compatible with stability (11) if f(x*) is sufficiently big, while p(x*) is sufficiently small. Turning to possibil- ity 2 (i.e., fx > jpxj > 0, and px < 0), we see that we cannot get stability be- cause the left-hand side of (11) will be positive, unless we admit negative functional responses. This shows that there are models with stable equilibria that violatetheVolterraPrinciple,notablymodelsthat violatetheVolterraPrin- ciple according to possibility 1. 4.4. Further Generalizations. Can the above results be generalized fur- ther? Theorem 1 depends on the scope of systems (3) and (4). This type of model is dubbed “intermediate” by Freedman (1980). There is a lot of space for generalization between Gause-type models and fully general dynamical 17. If we use a function q(x) instead of p(x) in the predator eq. (4) in the model, there are more possibilities for violating the Volterra Principle—a decreasing q, combined with increasing p and decreasing f, yields a violation of the Volterra Principle. However, it is unclear how realistic it is to assume that the functions p and q show a qualitatively different behavior. 750 TIM RÄZ systems.18 Here I want to mention a different kind of intermediate model that is important in population ecology. Systems (3) and (4) have so-called laissez-faire dynamics; that is, the functional response (which is also the nu- merical response) p(x) does not depend on the predator abundance y. Some population ecologists believe that this is problematic because these models do not allow for nontrivial predator interference.19 To overcome the prob- lems of laissez-faire dynamics, Arditi and Ginzburg (1989) have proposed so-called ratio-dependent predator-prey models in which the functional and numerical responses are functions of the ratio between predator and prey abundances. Ratio-dependent models have been hotly debated in population ecology in recent decades.20 It would be interesting to see whether the Vol- terra Principle can be extended to this type of model as well. 5. Weisberg’s Volterra Principle Revisited. In this section, I discuss the ramifications of the results in section 4 for Weisberg and Reisman’s work on the Volterra Principle, as far as the scientific and mathematical substance of their contribution is concerned. First, the fact that the Volterra Principle holds for models with density dependence and predator satiation, which was pointed out by Weisberg, is implied by theorem 1: one merely has to observe that these models are special cases of (3) and (4) (see the appendix). What is more, theorem 1 not only shows that these two models satisfy the Volterra Principle but also shows that this is due to the form of the growth function f and of the functional and the numerical response p. I turn to the ramifications of these observations for robustness analysis below. Second, the analysis of the scope of theorem 1, provided in section 4.3, shows that VP, the formulationof the Volterra Principle in Weisberg and Reis- man (2008), is incorrect. This is particularly clear if we consider possibility 1, according to which a violation of the Volterra Principle can occur if the growth function f is increasing. However, f is independent of negative coupling, which only concerns the interaction terms. Therefore, the Volterra Principle can be violated if there is negative coupling, and VP is wrong. Third, the equilibrium of a predator-prey system need not necessarily be the (average) abundance of that system, as I noted in section 4.2 above. Weis- 18. A general approach to predator-prey models is via so-called Kolmogorov-type mod- els (see, e.g., Freedman 1980, chap. 5). This approach was first proposed by Kolmogorov (1978). Kolmogorov-type models take general dynamical systems of two variables as a starting point and impose certain restrictions on these systems in order to obtain predator- prey relationships. 19. See Yodzis (1994) for a discussion of the notion of laissez-faire dynamics and rea- sons for taking predator interference into account. 20. See Arditi and Ginzburg (2012) for a book-length discussion of ratio-dependent models and Arditi and Ginzburg (2014) and Barraquand (2014) for discussion. VOLTERRA PRINCIPLE GENERALIZED 751 berg and Reisman do not explicitly address this problem. I already pointed out that Weisberg (2006b, 736) proposes extending the Volterra Principle to a model with predator satiation (see the appendix). This model satisfies the Volterra Principle, according to theorem 1. However, the model exhibits limit cycles. An argument analogous to Volterra’s Second Law would be nec- essary to show that the equilibrium of this system is relevant to the average abundance; without such an argument, the Volterra Principle may be ir- relevant. There is no such argument in Weisberg (2006b). To be fair, Weis- berg and Reisman (2008, 119) implicitly acknowledge that the stability of equilibria is relevant to the Volterra Principle. They compare the significance of equilibria, in the case of the original Lotka-Volterra predator-prey model, one the one hand, and in the case of the model with density dependent prey, on the other, and note that in the former case, the equilibrium is the average abundance, while in the latter case, the system will actually converge toward the equilibrium. They seem to endorse the system being at equilibrium (on average) as a necessary condition for the Volterra Principle’s being biolog- ically meaningful. 6. Generalization and Robustness Analysis. In this section, I discuss the upshot of the results in section 4 for robustness analysis, with a focus on Weisberg’s (2006b) account of robustness analysis. In section 4, I introduced one general model, the Gause-type model, which encompasses all the differ- ent models from Weisberg (2006b). I then showed that the Gause-type model has the Volterra Property. This generalization shows that the models investi- gated by Weisberg are not really independent, but rather belong to the same type, and that they all satisfy the Volterra Principle because they are of this type. Generalization and robustness analysis are similar in that they have the same goal. Both try to establish that a certain result, such as the Volterra Prin- ciple, does not depend on some of the idealizations of a particular model, such as the original Lotka-Volterra model. However, generalization achieves this goal on a path different from robustness analysis: it establishes that cer- tain idealizations of the original model are not necessary and that weaker assumptions suffice because we can derive the same result from one more general model instead of using multiple independent models. Note that gen- eralization does not get rid of all idealizations: The generalization of a result is only as good as the idealizations used in the general model. However, if we generalize a result, we get rid of some idealizations, that is, the assumptions dropped in the more general model. Furthermore, generalization can be su- perior to robustness analysis in that it may use less idealizations. In the cur- rent case, the Gause-type model is much more general than all the models used by Weisberg taken together, and thus it is preferable from the point of view of idealizations. 752 TIM RÄZ There are further important differences between the two accounts. First of all, the scope of generalization is more limited than that of robustness anal- ysis. Weisberg anticipates this point. In the following passage, he comments on a possible outcome of robustness analysis that is akin to generalization: In straightforward cases, the common structure is literally the same math- ematical structure in each model. In such cases one can isolate the common structure and, using mathematical analysis, verify the fact that the common structure gives rise to the robust property. However, such a procedure is not always possible because models may be developed in different mathemat- ical frameworks or may represent a similar causal structure in different ways or at different levels of abstraction. Such cases are much harder to de- scribe in general, relying as they do on the theorist’s ability to judge rele- vantly similar structures. (Weisberg 2006b, 738) The generalization approach pursued in section 4 is such a “straightfor- ward case.” The Volterra Principle turned out to be an actual theorem for the Gause-type model. I agree with Weisberg that such an outcome should not be expected in all cases—generalization has its limits. However, we can be more specific about the kinds of situations in which generalization may be applicable. I now turn to this task. The generalization approach will usually not work if models are from dif- ferent mathematical frameworks, as Weisberg notes. The results in section 4 concern models that are ordinary differential equations. Their robustness is thus restricted to this particular mathematical framework and does not extend to, say, IBMs, as discussed in Weisberg and Reisman (2008). Put dif- ferently, the generalization approach does not work in the case of repre- sentational robustness (i.e., when models belong to different mathematical frameworks): if two models belong to different frameworks, they may not share a common mathematical structure, and it is hard, if not impossible, to treat them in a formally unified manner. The generalization approach is usually restricted to cases of structural robustness (i.e., to models from the same mathematical framework). If we want to apply the generalization approach, it is not sufficient that the models are formulated within the same mathematical framework. It can be impossible to find a common generalization of models, even if they are for- mulated in the same mathematical framework. For example, Volterra (1978) explores properties of population fluctuations; the third property, the “distur- bance of averages,” is the Volterra Property. In part 2, paragraph 5, Volterra generalizes the Volterra Principle to systems with n species. Volterra notes that, in order for this to make sense, one has to restrict n-species models in several ways. For example, if one wants to be able to compare the abun- VOLTERRA PRINCIPLE GENERALIZED 753 dances of predators and prey, one needs an absolute distinction between pred- ators and prey. Cases in which one species is preyed on by a second species, while also being the predator of a third species, have to be excluded because it is not clear how the Volterra Principle should even be formulated in such cases. However, Volterra shows that, under certain technical assumptions, a gen- eralized analogue of the Volterra Principle can be proven if the system con- tains an equal number of predator and prey species. This draws attention to the fact that the precise formulation of the property that is expected to be ro- bust is important for generalization. In the cases examined by Volterra, the ro- bust property is not the same for two species and for n-species systems. In the former case, the property is formulated in terms of the ratio of two species, whereas in the latter case, the property concerns the ratio of (equally many) predator and prey species. Thus, the generalization approach can be difficult to carry out, if the robust property is not the same in all cases and needs to be reformulated. In the case of the models examined in Weisberg (2006b), which served as the starting point of generalization, we find that the robust property in ques- tion, the Volterra Property, is the same formal statement in all cases. This is not an assumption that is generally made in robustness analysis. Generally, a robust property can be characterized informally, and the statements are only required to be similar, or “biologically the same.” A second commonality of the models examined in Weisberg (2006b) is that the derivations of the Volterra Property, while different, are similar. In sum, I suggest that the generalization approach should be applied as a rule of thumb in the following situation: if a series of models, formulated in the same mathematical framework, yields identical formulations of a robust property, and the derivations proceed in a similar manner, we should try to identify one general model from which we can derive the robust property. In this kind of situation, robustness analysis serves as a heuristic strategy: the point of deriving the same result from a range of different models serves the ultimate goal of finding a general result by examining some instances first. 7. The Generalized Volterra Principle as an Explanation. The Volterra Principle was originally proposed as an explanation of an empirical regular- ity, as I pointed out in section 2 above. It is, therefore, natural to ask whether the results in section 4 are explanatorily significant. I discuss this issue in the current section. I argue that the results in section 4 exhibit several explana- tory virtues that have been proposed in the debate on mathematical explana- tions. Before I discuss the explanatory virtues, some clarifications are in order. The original explanandum is an empirical regularity observed in fishery sta- tistics. These findings have been disputed on empirical and statistical grounds 754 TIM RÄZ (e.g., by Pearson 1927). Of course, the facticity of the explanandum is nec- essary if we want to give a real explanation.21 Here I do not discuss empirical questions, but I rather assume that we are dealing with an actual empirical phenomenon, and I assess the theoretical virtues of the results in section 4 under this assumption. The explanation I discuss is that the Volterra Property in Gause-type models is explained by the fact that the growth rate function f is decreasing, while the functional response p is increasing. One of the explanatory virtues of the above results consists in their gen- erality, or unificatory power, and we can spell this out in terms of unification, as proposed by Kitcher (1989): if we compare Weisberg’s results with those given here, we see that the explanation I propose is more unified and that it subsumes different, but similar, arguments given by Weisberg, under one ar- gument scheme, in a natural way. Now, it has been argued in the literature on mathematical explanations that unification, and generality, cannot be all there is to mathematical explanation.22 I agree that while generality and uni- fication do contribute to explanatory power, they are not all there is to it. A second explanatory virtue of the results in section 4 is that they estab- lish natural boundaries for the validity of the Volterra Principle.23 I showed that the Volterra Principle is not valid if the assumptions on the functions f and p are weakened any further. These properties are minimal, in that if the growth rate f increases, the explanandum does not follow; similarly, the equilibrium may become unstable, if the functional response p decreases. These properties are not only sufficient but also necessary for the Volterra Principle, against the background of Gause-type models. This fits nicely with a recent proposal by Pincock (2015), who requires conditions that are nec- essary as well as sufficient for “abstract explanations”: a kind of mathemat- ical explanation. Pincock’s idea is that requiring the condition to be neces- sary as well as sufficient makes sure that there are no redundant assumptions in the explanans, in that if the assumptions are weakened, the explanation fails. This is exactly what we see here. A third explanatory virtue of the Gause-type explanation emerges, if we analyze its relation to the other less general explanations. Volterra’s explana- tion, based on the original predator-prey model, establishes that it is possible to account for the Volterra Property on the basis of systems of coupled dif- ferential equations. Weisberg showed that Volterra’s result can be extended 21. Additionally, one should always compare the mechanism producing the empirical phenomenon with the model used in the explanation, in order to assess the explanation’s adequacy. 22. See Hafner and Mancosu (2008) for a critical discussion of unification à la Kitcher in the context of mathematical explanation and Steiner (1978) for arguments against a characterization of explanatory power in terms of generality. 23. The importance of establishing the limits of a result to robustness analysis has been noted by Wimsatt (2012b). VOLTERRA PRINCIPLE GENERALIZED 755 to other more realistic models. This establishes that the explanation is not an artifact of the idealizations of the original predator-prey model. What the ex- planation based on the Gause-type model achieves is the identification of the properties that are ultimately responsible for the explanandum, that is, the (mathematical) Volterra Property. By doing this, the Gause-type explanation not only accounts for the explanandum, but it also explains why the other explanations, proposed by Volterra and Weisberg respectively, worked in the first place: the growth function and the functional response of all these explanations have the right properties. The idea that explanatory relations of this kind, between different explanations, can be interpreted as an explan- atoryvirtuehas recently been proposed byLange (2015). Accordingto Lange, a mathematical result that answers questions that other results leave open is “deep” and has high explanatory power. This is a further important feature of the results presented here. In sum, the results regarding the Volterra Principle presented above, have several explanatory virtues. They are general, they provide an explanans that is sufficient and necessary in the case of Gause-type models, and they are “deep,” in explaining why previous explanations worked. 8. Robustness Analysis and Coincidence. While it is desirable to apply the generalization approach, there is no guarantee that it will always work. What if the generalization approach yields no result? In the current section, I address this question. I will argue that in cases in which generalization can- not be applied, robustness analysis and explanation pull in different direc- tions. I restrict attention to cases in which generalization is a promising ap- proach in principle, that is, cases in which the same mathematical statement is derived from multiple independent models within the same mathematical framework, as discussed in section 6. Consider, for a moment, the reasoning process of a theorist who wants to establish the robustness of a property, such as the Volterra Property. The the- orist examines a range of different models within the same mathematical framework and succeeds in deriving the same property from different mod- els. In this situation, the theorist can be confident that the property is in fact robust. However, the theorist will not stop there. She will also wonder whether the fact that she succeeded in deriving the result in such a way has a common explanation. In fact, there is a certain urgency to find such an explanation be- cause this would account for the fact that multiple derivations are possible. If, however, no such explanation is to be had, what does this tell us? It may tell us that the existence of multiple derivations is a mere coincidence. It could be questioned whether mathematical coincidences are possible at all, given that all mathematical truths are necessary. At the very least, the no- tion of a mathematical coincidence needs to be clarified. The notion has been discussed by Lange (2010). Lange gives the following example of a math- 756 TIM RÄZ ematical coincidence: the thirteenth digit of the decimal expansion of both e and p is 9. There is no reason, or common explanation, for this fact; at least, no such explanation is known. It is a mere coincidence. In a nutshell, Lange’s account of mathematical coincidence is that two mathematical facts F and G are coincidences, if there is no common mathematical explanation of F and G. If we want this definition to be nontrivial, we have to make some requirements on the “naturalness” that a common mathematical explanation would have to satisfy; otherwise, a mere conjunction of any two explana- tions would be sufficient. One of the main challenges is to spell out just what such a requirement of “naturalness” amounts to. I will not discuss this is- sue in the abstract but turn to the case at hand instead. Lange’s idea applies to the case of the Volterra Principle, as follows. The mathematical coincidence is that we can derive the Volterra Property from the original predator-prey model, as well as from both the model with den- sity dependence and the model with predator satiation; all of this was es- tablished in Weisberg (2006b). This situation, the (apparent) coincidence, prompts the request for an explanation. The explanation is provided in sec- tion 4 above: the different models are special cases of the Gause-type model, from which the Volterra Property can be derived.24 Thus, in some cases, the process of robustness analysis can itself prompt a request for an explana- tion, and in a subset of these cases—including the generalized Volterra Prin- ciple—an explanation can be provided, and the apparent coincidence is ex- plained away. What about real coincidences? In these cases, we succeed in deriving the same property from multiple models, but there is no common explanation of the fact that these multiple derivations were possible. This fact is a mere co- incidence. In this case, the following situation arises. Assume that we want to check whether some idealizations in a model M are responsible for the explanation of a property P or whether the explanation is independent of these idealizations. According to Schupbach (2016), we can do this using ro- bustness analysis. Assume, further, that we construct two models, M1 and M2; that, in these two models, different idealizations from M are removed; and that we succeed in deriving the property P from M1 and M2. Assume, fi- nally, that this is a mathematical coincidence. This means that there is no ex- planation as to why we succeeded in deriving the property P from M1 and M2. If this is the case, we will not be able to construct a model M 0 that en- compasses M1 and M2, and from which we can derive P, insofar as such a derivation constitutes an explanation telling us why the multiple indepen- 24. I am not suggesting here that all mathematical explanations have the form of gener- alizations or unification; I merely claim that the results in sec. 4 do in fact explain the (ap- parent) coincidence in Weisberg (2006b), i.e., that the Volterra Property can be derived from multiple models. VOLTERRA PRINCIPLE GENERALIZED 757 dent derivations are possible. This, in turn, means that the different idealiza- tions in M, which were removed in M1 and M2 individually, cannot be re- moved collectively by constructing one model M 0, such that M 0 would ex- plain P. This is worrisome because having at least one of the idealizations seems to be essential to the derivation of the property P: it is impossible to remove all the idealizations in the two models at the same time by construct- ing an explanatory model. Note that the explanatory goal of robustness anal- ysis, according to Schupbach’s proposal, can still be achieved to a certain de- gree. We can control for some idealizations in a model, by deriving a property from different models, thus ruling out these idealizations as explanations. However, some idealization cannot be removed from our explanation. In sum, it has to be expected that generalization will not work in all cases. The interpretation of such cases as mathematical coincidences has interest- ing consequences, in particular the impossibility of finding a general explan- atory model that removes all idealizations we would like to control for. This does not mean that generalization or mathematical coincidences speak against robustness analysis. Rather, we should interpret robustness analysis as a heu- ristic strategy that can lead to several possible outcomes. On the one hand, we may discover a general model that provides a unified explanation of a mathematical fact. On the other hand, we may find that a mathematical fact that can be derived from several independent models is merely an example of a mathematical coincidence. In the latter case, the outcome of robustness analysis is that an explanation is not to be had and that some idealizations are here to stay. 9. Conclusion. In the current article, I have shown that the Volterra Princi- ple can be generalized for Gause-type models, extending results by Weis- berg and Reisman. I have argued that this generalization has some of the virtues of mathematical explanations; in particular, the limits of the results establish necessary and sufficient conditions, and the results illuminate pre- vious explanations. I have also proposed that cases in which the generaliza- tion approach is not applicable can be interpreted as mathematical coinci- dences. The current article reveals a close connection between the goals of robust- ness analysis and those of purely mathematical work on theoretical models. Generalizing existing results is a very common practice in mathematics. How- ever, there are other aspects of mathematics that should also be explored in the context of robustness analysis. First, more should be said about cases of representational robustness, where the robust results stem from different mathematical frameworks. These are cases of analogical reasoning in math- ematics, the status of which is all but clear. Second, more complicated cases of structural robustness have scarcely been discussed so far. For example, a systematic study of “analogical generalizations,” such as Volterra’s general- 758 TIM RÄZ izations of the Volterra Principle to n-species systems, would be worthwhile. Third, the results in the current article could also be further generalized to other kinds of predator-prey models. Appendix Models with Density Dependence and Predator Satiation After normalizing parameters a and b, the model with density dependent prey (Weisberg 2006b, eqq. [6] and [7]) can be written as _x 5 xr 1 2 x K � � 2 xy: (A1) _y 5 xy 2 my: (A2) In this model, the first summand in equation (1) is modified such that the prey population does not grow indefinitely in the absence of predators, but rather the growth is limited by a maximal number of prey, the carrying capacity K. This model can be seen to be of Gause type by letting f (x) 5 1 2 (x=K) and p(x) 5 x. The model with predator satiation (Weisberg 2006b, eqq. [9] and [10]), after normalizing parameters a, b, and c, can be written as _x 5 xr 1 2 x K � � 2 1 2 e2xð Þy: (A3) _y 5 1 2 e2xð Þy 2 my: (A4) This model takes into account that the rate at which predators capture prey may not be constant, as in the Lotka-Volterra model, but may rather decrease with increasing prey density. This model can be seen to be of Gause type, by choosing f (x) 5 1 2 (x=K ) and p(x) 5 1 2 e2x. REFERENCES Arditi, R., and L. R. 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