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Journal of Theoretical Biology 310 (2012) 31–42 Contents lists available at SciVerse ScienceDirect Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi Optimal nutrient foraging strategy of an omnivore: Liebig’s law determining numerical response József Garay a,n, Zoltán Varga b, Tomás Cabello c, Manuel Gámez d a Research Group of Theoretical Biology and Evolutionary Ecology of the Hungarian Academy of Sciences and Department of Plant Systematics, Ecology and Theoretical Biology, L. Eötvös University, Pázmány P. sétány 1/C, H-1117 Budapest, Hungary b + Hungary Institute of Mathematics and Informatics, Szent István University, Páter K. u. 1., H-2103 Gödöllo, c Department of Applied Biology, University of Almerı́a, Ctra. Sacramento s/n, ES-04120 Almeria, Spain d Department of Statistics and Applied Mathematics, University of Almerı́a, La Cañada de San Urbano s/n, 04120 Almeria, Spain H I G H L I G H T S c c c c We build a bridge between stoichiometric ecology and optimal foraging models. Give a theoretical explanation of a bistable coexistence observed in omnivory. Our model includes different biological mechanisms at a time. May find application in nutritional ecology, intraguild predation and pest control. a r t i c l e i n f o abstract Article history: Received 5 May 2012 Received in revised form 14 June 2012 Accepted 15 June 2012 Available online 28 June 2012 The paper is aimed at a theoretical explanation of the following phenomenon. In biological pest control in greenhouses, if an omnivore agent is released before the arrival of the pest, the agent may be able to colonize, feeding only on plant and then control its arriving prey to a low density. If the pest arrives before the release of the agent, then it tends to reach a high density, in spite of the action of the agent. This means that according to the initial state, the system displays different stable equilibria, i.e. bistable coexistence is observed. Based on the biological situation, the explaining theoretical model must take into account the stoichiometry of different nutrients and the optimal foraging of the omnivore agent. We introduce an optimal numerical response which depends on the optimal functional responses and on the ‘mixed diet–fitness’ correspondence determined by ‘egg stoichiometry’, in our case by Liebig’s Law; moreover we also study the dynamical consequences of the latter when the plant is ‘‘inexhaustible’’. In our model, we found that under Holling type II functional response, the omnivore–prey system has a unique equilibrium, while for Holling type III, we obtained bistable coexistence. The latter fact also explains the above phenomenon that an omnivore agent may control the pest to different levels, according to the timing of the release of the agent. & 2012 Elsevier Ltd. All rights reserved. Keywords: Ecological stoichiometry Imperfectly substitutable resources Liebig’s Law Numerical response Omnivory 1. Inroduction 1.1. The theoretical problem The basic experimental motivation of our present study is a phenomenon observed in greenhouses, where an omnivore agent feeding only on a plant, reached a lower population density than in case of living on a mixed diet. If this agent is released before the arrival of the pest, the agent may be able to colonize and then control the arriving pest to a low density. If the pest arrives before the release of the agent, then it tends to reach a high equilibrium density, in spite of the action of the agent. At theoretical level this means that according to the initial state, the system displays different stable equilibria, i.e. bistable coexistence is observed. In greenhouse tomato crops, for instance, the phytophagous spider mites (Tetranychus urticae Kock) (Acari: Prostigmata: Tetranychidae) and its omnivore predator – Phytoseiulus persimilis AthiasHenriot (Acari: Mesostigmata: Phytoseiidae), also feeding on pollen – are a good example for the above phenomenon (Gerson et al., 2003). In the considered biological situation, for a general theoretical model, the following three mechanisms must be taken into account: stoichiometry, optimal foraging and population dynamics based on functional response. n Corresponding author. Tel.: þ36 1 3722500/8798; fax: þ36 1 3812188. E-mail addresses: [email protected] (J. Garay), [email protected] (Z. Varga), [email protected] (T. Cabello), [email protected] (M. Gámez). 0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.06.021 1.1.1. Stoichiometry In many cases mixed diet can improve the fitness of consumers (see e.g. Raubenheimer et al., 2009; Groenteman et al., 2006; Bilde 32 J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42 and Toft, 1994). For instance, Shakya et al. (2009) found that intraguild predation (IGP) was more intensive on pollen-breeding than pollen-free flowers and the plant food (flowers, fruit, leaves) was also important. Of course, the nutrient compositions of plant and insect diet items are very complex (Romeis et al., 2005), but it is observed in omnivore predators that when the usual prey diet is supplemented with plant food, their development rate generally increases and so do the other biological parameters such as survive rate, longevity and/or fecundity of adults (Eubanks and Styrsky, 2005). These observations confirm that, the food intake has a strong effect on the growth rate of omnivore and its prey. The importance of stoichiometry was emphasized by Simpson et al. (2004) in general: ‘‘There are growing evidences that, rather than maximizing energy intake subject to constraints, many animals attempt to regulate intake of multiple nutrients independently.’’ A well-known example for that is the pollen collection by bumblebees (e.g. Rasheed and Harger, 1997). 1.1.2. Optimal behavior The consumer’s foraging strategy will depend on its stoichiometric constraints, (cf. Shakya et al. 2009). Moreover, Eubanks and Denno (2000) found that the presence of high-quality plant parts reduce the number of prey consumed by omnivore. 1.1.3. Population dynamics The consumer foraging strategy depending on its stoichiometric constraints has an important effect on the dynamics of the interacting species, e.g. van Rijn et al. (2002) found that addition of pollen to a young mature leaf of male sterile cucumber plant increased omnivore predator population growth and reduced herbivore numbers. Clearly, the functional response also plays a crucial role in the population dynamics of the system. A basic aim of this paper is to give a theoretical explanation of the phenomenon recalled at the beginning of Section 1, taking into account the above three mechanisms. 1.2. Theoretical preliminaries: why must we build up a new model? In classical optimal foraging theory, in most cases, the energy intake is optimized (e.g. Stephens and Krebs, 1986). However, the nutrient contents of the food may also be an important issue, see e.g. León and Tumpson (1975) where, borrowed from classical microeconomics, the concepts of ‘‘perfectly substitutable’’ resources (where the difference between them is only quantitative) and ‘‘perfectly complementary’’ resources (where each one fulfils one of the consumer’s essential needs) have been introduced. Clearly, the nutrition compositions of two resources have an effect on the coexistence of two consumers (Abrams, 1987a). This question also was studied from evolutionary perspective: Abrams (1987b, c) pointed out that when the resources are perfectly substitutable or complementary, divergence away from or convergence towards the competitor’s resource-acquisition traits happens on evolutionary time scale. Furthermore, Fox and Vasseur (2008) also found that competition generates convergence. Surprisingly, in optimal foraging theory and population dynamics this topic was a little neglected, except the following two basic papers: Abrams (1987d), starting from the terminology of León and Tumpson (1975), gave a general overview of adaptive functional responses. Recently, Vasseur and Fox (2011) developed Abrams’s setup, studying its evolutionary consequences. In the present paper, we also elaborate Abrams’s concept, calculate the optimal foraging strategy (determining optimal consumption rates) and study the population dynamic consequence of it. As a matter of fact, bistable coexistence under nutrition constraints has not been found in the above theoretical models. According to our motivating phenomenon in which omnivore can live feeding only on plant and also feeding only on prey, we have to consider ‘‘imperfectly substitutable resources, on any of which alone the consumer can subsist, but if taken together would improve the consumer fitness’’ (León and Tumpson, 1975). Throughout the paper we will assume that the different foods contain enough energy for the consumers to survive, thus a consumer is only interested in nutrients that are essential for its fitness. We claim, that in the existing theoretical studies bistable coexistence has not been observed, since they did not consider imperfectly substitutable resources with Libieg’s Law and have not considered Holling III functional response, at the same time. 1.3. Basic assumptions of the present work Based on the above points, for the modeling of the motivating phenomenon the above three mechanisms are indispensable according to the greenhouse observations. But from the point of view of applied ecology, neither stoichiometry nor optimal foraging of the agent can be omitted; we could only vary the functional response. What we will demonstrate is that while in case of Holling type II there was no bistable coexistence, in case of Holling type III there was. During our model building we will focus on the nutrient composition of different diet items, based on the fact that there are some experiments in which the food intake of the IG predator does not maximize its energy income (Hailey et al., 1998), so our approach is motivated by stoichiometrical ecological dynamics (Andersen et al., 2004), in the sense that the IG predator growth rate depends on the total intake of different nutrients from the consumed food items. This view is in harmony with the observation that the relative abundance of nitrogen and phosphor in biomass is closely associated in insects (Elser et al., 2003); for nitrogen and carbon, see Denno and Fagan (2003). We will consider a system, where plant and prey are imperfectly substitutable resources, thus we will consider the case when each food item is incomplete at least for one nutrient, thus the omnivore predator has no unique optimal food item for reproduction. We assume that stoichiometric constraints can be described by Liebig’s Law, which determines the optimal foraging strategy of the omnivore. This idea is basically different from the usual approach of optimal foraging theory, which assumes that according to the densities of different foods there is always only one which is optimal (e.g. Křivan, 2000; Křivan and Sikder, 1999), and it is usually assumed that for an omnivore it is only the energy intake that counts, and there is no nutrient constraint. We assume both food items are rich in energy and some nutrients are needed for the survival of omnivore adults, thus the nutrient constraints determine the fitness of the omnivore (cf. Pulliam, 1975). More precisely, we assume that only the fitness depends on the mixed diet, and the survival probability of adults does not.1 In this paper we will consider the simplest theoretical situation, assuming that there is only one plant species, which produces food, say pollen, for the omnivore; while the herbivorous prey feeds on the leaves of the plant. The plant food is not exhausted2 by the 1 Moreover, we ignore the associative effect of different diet items, i.e. when one diet item affects the digestion of another diet item (e.g. Bjorndal, 1991). Although we are aware that the toxin dilution has an important rule in diet mixing (e.g. Hirakawa, 1995; Singer et al., 2002); for the sake of simplicity, we assume each food of the omnivore contains no toxin. This is particularly true in cultivated plants, due to the loss of many of the above mentioned substances in the process of domestication and transformation in agronomic varieties (Kogan, 1994). 2 This simplifying assumption is also used by Coll (2009), Srinivasu and Prasad (2010a, b) and van Rijn and Sabelies (2005). J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42 omnivore species, and herbivores do not feed on pollen. According to our simplifying assumption we will not model the change of plant biomass, we will assume there is always a fixed quantity of pollen, the overflowing pollen falls to the ground.3 In other words, we assume that plant food provided4 by the plant to the omnivore is a non-exhaustible resource and there is no competition for the plant food between the omnivore and its prey. Thus we will not investigate the coexistence of the predator–consumer-plant IGP system (e.g. Holt and Huxel, 2007; Holt and Polis, 1997; Křivan, 2000; Křivan and Diehl, 2005). We consider only the dynamics of two interacting species: omnivore as predator and herbivore as prey; omnivore has two kinds of food: pollen and prey.5 Our model building consists of the following steps: First step: using a new method (see Appendix), for a fixed (arbitrary) foraging strategy of the omnivore, we derive the functional response corresponding to fixed densities of different food items. Second step: we calculate the nutrition intakes determined by the derived functional response. Third step: according to the mixed diet–fitness function, we calculate the optimal foraging strategy corresponding to the nutrient constraints. Observe that in this way we actually calculate the (optimal) numerical response. Fourth step: we consider the dynamic consequence of the fact that functional and numerical responses are connected by stoichiometric contraints. In Section 2, for the case of Holling type II functional response, the mixed diet–fitness mapping for the omnivore and the optimal foraging strategy of omnivore under nutrition constraint are calculated. As a consequence, the population dynamics corresponding to the mixed diet of the omnivore is presented, which implies unique stable coexistence. In Section 3, it is shown that the considered ecological conditions lead to Holling type III functional response, and for the corresponding dynamics bistable coexistence is obtained. Sections 4 and 5 present a summary of our results and their discussion with an outlook, respectively. 2. Case of a Holling type II functional response In this section we will concentrate on the fact that the increasing prey density must increase the number of satiated omnivore predators. 2.1. Calculation of the functional response For the derivation of the functional response, we have to determine how the activity probabilities pi depend on the pollen and prey densities. Let us denote by sA[0,1] the time proportion dedicated to pollen by the omnivore, considering its foraging strategy, constant for a short time period. Now our basic assumption is that the food intake of the omnivore linearly depends on both the density of prey and the pollen supply. Let p be the probability that there is pollen in a perception range of the omnivore. If the prey density (biomass) is y and N is the number of perception ranges 33 covering the considered area, the probability that there is a prey in the perception range of the omnivore then is y/N. Given a perception range, suppose that for the omnivore k1 and k2 are the probabilities to find pollen, and find and kill a prey, respectively. Then for the activity probabilities we have p1 :¼ sð1k1 Þp þsð1pÞ ¼ s 1k1 p , y y y ¼ ð1sÞ 1k2 , p2 :¼ ð1sÞð1k2 Þ þ ð1sÞ 1 N N N p3 :¼ sk1 p, y p4 :¼ ð1sÞk2 : N Thus the average time dedicated to the considered activities is y EðTÞ ¼ T 1 þ ðT 3 T 1 Þk1 pT 2 ðT 4 T 2 Þk2 ðy=NÞ s þ T 2 þ ðT 4 T 2 Þk2 : N So we have the following functional responses of the omnivore to pollen and prey, respectively: Pðs,yÞ :¼ sk1 p EðTÞ and Fðs,yÞ :¼ ð1sÞk2 ðy=NÞ : EðTÞ We remind that s ¼1 means that the omnivore feeds only on pollen, and in case of s ¼0 it feeds only on prey. Observe that at fixed foraging strategy of the omnivore we get a Holling II type functional response. 2.2. Nutrient intake and mixed diet–fitness mapping For the sake of simplicity and more transparency, we minimize our model, considering two chemical elements, say nitrogen and carbon as nutrients, that determine the fitness of the omnivore. This choice is only illustrative, since by its importance, e.g. phosphorus can be also considered, see Perkins et al. (2004) Acharya et al. (2004). We can assume the unit of each nutrient is its quantity contained in an omnivore’s egg.6 Let ðn1 =c1 Þ and ðn2 =c2 Þ be the nutrient intake vectors resulting from the consumption of pollen and prey, respectively. (For example, n1 and c1 are the nitrogen and carbon absorbed from the consumption of a unit quantity of pollen, respectively.) For the fixed (s,y), the total nutrient intake function is ! ! n1 pk1 s þ n2 k2 Ny ð1sÞ n1 Pðs,yÞ þ n2 Fðs,yÞ 1 y Mðs,yÞ :¼ : ¼ c1 Pðs,yÞ þ c2 Fðs,yÞ EðTÞ c1 pk1 s þ c2 k2 N ð1sÞ Now assume that pollen is abundant in the sense that p does not depend on the consumption rate of pollen. According to Liebig’s Law (Huisman and Weissing, 1999), the fitness (i.e. the biomass of produced eggs) of the omnivore is determined by the minimum coordinate of nutrient intake function M(s,y): min M i ðs,yÞ ¼ min n1 Pðs,yÞ þ n2 Fðs,yÞ,c1 Pðs,yÞ þ c2 Fðs,yÞ : i 3 In greenhouse crops (not in open air crops), which are closed and small spaces with relatively low density of flowers, the intensive use of pollinator (bumblebees or honeybees) can reduce the pollen available to the omnivore. 4 Relatively few theoretical studies have addressed the impact of plantprovided food on herbivore–carnivore dynamics, for a review, see van Rijn and Sabelies (2005). 5 In our simplest situation, we can ignore other, non-nutritive factors, which have effect on the diet mix (e.g. the omnivore may also be exposed to predation risk in certain regions of its habitat, see Hailey et al. (1998) and Singer and Bernays (2003). Finally, we also ignore the stage structure of the species, in other words we do not deal with the dynamics of different stages of development of insects (van Rijn et al., 2002). We also ignored the stage structure of the plant, because in the particular conditions of greenhouses crops, the plants are transplanted from plant nurseries after 1 or 2 months with more than 6 leaves, and then they start to flower almost immediately. Remark. Liebig’s Law allows us to consider in the same model the following three cases, according to the terminology of León and Tumpson (1975): 1. Perfectly substitutable resources that satisfy the same requisite needs, e.g. n1 ¼n2 and c1 ¼c2; 2. Perfectly 6 Although all insects require the essential amino acids (methionine, threonine etc.), many of them can synthesize amino acids (Lundgren, 2009). Instead of that, we only think of chemical elements (e.g. nitrogen and carbon) contained in pollen and prey, since these are clearly essential. Of course, chemical elements cannot be decomposed by any organism, even some carbohydrates (e.g. starch and cellulose) are also not digestible by certain insects, see Cohen (2004). 34 J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42 complementary resources, where each of them fulfils one of the consumer’s essential needs, e.g. n1 a0, n2 40 and c1 40, c2 ¼0. 3. Imperfectly substitutable resources, on any of which alone the consumer can subsist, but which if taken together would improve the consumer fitness, e.g. n1 on2 and c1 4c2. In cases 1 and 3, the consumer can grow on a single resource. In the case of perfectly substitutable resources, fitness optimization means maximization of the biomass intake, which is similar to the standard idea of optimal foraging theory. 2.3. Optimal foraging strategy of omnivore under nutrition constraint Assume that there is no optimal food for omnivore, i.e. n1 on2 and c1 4c2. Now the optimal foraging strategy depends on two different factors: the searching and handling times, and the nutrition intakes from different food items. We will consider two situations. 1. Prey is scarce, thus the corresponding average searching time is too long in the sense that the omnivore is better off by looking for pollen. 2. Prey population is large enough, so for the omnivore it is remunerative to look for prey. 2.3.1. Switching prey density When does omnivore start to feed on its prey? When y is very small, nitrogen is scarce for the omnivore, thus the fitness of omnivore is determined by n(s,y). Suppose that prey is very rare, yE0. Then h ¼0, and there is a critical level y0 for the prey density, below which the omnivore would not look for prey, since the expected intakes from insects are too low compared with the ‘‘cost’’ of searching. This is the case when in time unit the omnivore’s intake is greater, if it is looking only for pollen. Since, the omnivore wants to maximize its nitrogen-intake n1 k1 pn2 k2 ðy=NÞ s þ n2 k2 ðy=NÞ nðs,yÞ :¼ EðT S Þ by its optimal foraging strategy: an easy calculation shows that if y oy0 :¼ N n1 k1 pT 2 , n1 k1 pk2 ðT 2 T 4 Þ þ n2 k2 T 1 þ ðT 3 T 1 Þk1 p the omnivore would not eat prey. Here we assume that denominator is positive, which can be guaranteed by the simple assumption that the handling time of pollen, T4 T2 is shorter than that of insect, T3 T1.7 2.3.2. Optimal mixed diet Now assume that y 4y0. Then, according to Liebig’s Law the growth rate of omnivore is determined by the maximum of the minimum coordinate of nutrient intake function M(s,y). Actually, at the optimal foraging strategy denoted by s(y), both total nutrient intakes are equal. Indeed, if total intake of one nutrient is less than the other one, then the omnivore can increase its fitness by feeding more on the food containing more of the missing nutrient. If y4 y0, the omnivore maximizes min nðs,yÞ; cðs,yÞ at s(y). Let us denote c(s,y):¼c1P(s,y)þc2F(s,y), then at the optimal foraging strategy we have nðsðyÞ,yÞ ¼ cðsðyÞ,yÞ: The above equation yields sðyÞ ¼ k2 ðc2 n2 Þðy=NÞ : k1 ðn1 c1 Þp þ k2 ðc2 n2 Þðy=NÞ 7 If y0 o0, then there is no switching, since the prey is very rich in nitrogen. In this case, we have to consider opportunism as well, since when prey is very rare, the possible optimal behavior is this: when looking for abundant pollen, if a very rare prey is occasionally found, then consume prey. In order to simplify the mathematical discussion, we will not consider this case in the paper. Clearly, if c2 on2 and n1 oc1, then s(y)A]0,1[, so there is an optimal mixed diet of the omnivore. Example 1. For an illustration, we consider parameters n1 ¼5; c1 ¼10; n2 ¼20; c2 ¼ 9; k1 ¼0.06; k2 ¼0.46; p ¼ 0.9; T1 ¼T2 ¼1; T3 ¼6; T4 ¼11 and N ¼ 30. Now, the switching prey density is y0 ¼0.02, and for (s,y)A[0,1] [1,30] the total nitrogen and carbon intakes n(s,y) and c(s,y) are plotted in Fig. 1. Moreover, the intersection of the surfaces of Fig. 1 represents the optimal foraging strategy s(y) which is also shown in Fig. 2. Even in case of several nutrients (e.g. carbon:nitrogen:phosphor see Vrede et al., 2004), according to Liebig’s Law, the optimal mixed diet is guaranteed by the maximum of the lower hull of the essential nutrient income functions. In other words: each foraging strategy determines several nutrient intakes, but according to Liebig’s Law, the fitness of omnivore is given by the minimal essential nutrient determined by its foraging strategy. Thus the consumer tends to maximize the minimal essential nutrient intake by changing its foraging strategy. Now let us assume that diet change of the omnivore is faster than the population density change (Krivan and Cressman, 2009). We need this condition for the application of the method of Garay and Móri (2010). Thus the functional responses determined by the optimal numerical response are the following: PðsðyÞ,yÞ :¼ k2 ðc2 n2 Þðy=NÞk1 p , T 1 1k1 p þ T 3 k1 p k2 ðc2 n2 Þðy=NÞ þ k1 ðn1 c1 Þp T 2 1k2 ðy=NÞ þ T 4 k2 ðy=NÞ FðsðyÞ,yÞ :¼ k1 ðn1 c1 Þpk2 ðy=NÞ , ½T 1 ð1k1 pÞ þ T 3 k1 pk2 ðc2 n2 Þðy=NÞ þ k1 ðn1 c1 Þp T 2 ð1k2 ðy=NÞÞ þ T 4 k2 ðy=NÞ Moreover, the numerical response determined by the optimal functional responses is the following: nðsðyÞ,yÞ ¼ n1 PðsðyÞ,yÞ þ n2 FðsðyÞ,yÞ: 2.4. Population dynamics when change in the optimal foraging is immediate Under our simplifying assumption that developmental stages of the omnivore are neglected and only the fitness depends on the mixed diet, and the survival probability of adults does not; we can follow the standard dynamical setup well-known in the literature (Arditi et al., 2004; Danger et al., 2008; Miller et al., 2004), let us consider the following population dynamics: x_ ¼ xðmin nðs,yÞ; cðs,yÞ k1 xÞ Fðs,yÞ x : y_ ¼ y rk2 y y For y4y0, we obtain the optimal mixed diet dynamics: x_ ¼ xðnðsðyÞ,yÞk1 xÞ FðsðyÞ,yÞ x : y_ ¼ y rk2 y y Here the conversion efficiency for the predator is 1, since we assumed that the survival probability of adults does not depend on the diet mix, and the unit of each nutrient is equal to its quantity contained in an omnivore’s egg. An easy calculation shows that this system has at most two equilibria8, and at most one asymptotically stable equilibrium, where both species coexist, as illustrated in the following. 8 Using the index theory of planar vector fields, the following can be proved: for bistable coexistence the existence of at least three interior equilibria is necessary. J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42 35 Fig. 1. Total nitrogen and carbon intakes depending on two factors: density of prey (y) and foraging strategy of predator (s). Since the different nutrient intakes are measured in the nutrient contents of an egg of IG predator as a unit, the vertical axis corresponds to the fitness of IG predator. According to Liebig’s Law, the realized fitness of IG predator is determined by the minimum of both nutrient income functions, at each fixed prey density. Fig. 2. Optimal foraging strategy s(y) corresponding to prey density y. Observe that in our model, the optimal forging strategy is not a fixed ratio of different foods, and it does not follow the well-known one-zero rule of optimal foraging theory. Example 2. We consider the mixed diet dynamics with the parameters of Example 1, and r ¼3, k1 ¼0.46 and k2 ¼1. We have a positive equilibrium E0 ¼(1.32, 0.98), which is locally asymptotically as shown in Fig. 3. 3. Holling type III functional response: a two-patch model with density dependent perception The Holling type II functional response can take account of the fact that the increasing prey density must increase the number of satiated omnivore predators. However, when the density of prey is very small, there may be another effect, namely, the increase of the prey population implies a dramatic increase in the encounter rate, and at low prey density some satiation of the omnivores takes place, but it does not decrease dramatically the number of active omnivores. This interaction can be described by a Holling type III functional response. In this section we will follow the model-building scheme of the previous one for a different predation process, therefore some details will be skipped. 3.1. Functional response Let us consider the following simple two-patch model: Inside the flower: there is only pollen, and outside the flower there is prey. The omnivore diet strategy is the time h it spends outside the flower (then it spends time 1 h inside the flower). Since now the strategy of the IG predator is spacing we use a notation different from that of Section 2. In the flowers the pollen ‘‘density’’ is fixed, i.e. the omnivore finds pollen there with certain probability p0 (then it finds no pollen with probability 1 p0). Outside of flower: Assume the perception range contains only two home ranges of prey. Thus there are three types of perception ranges: empty, containing one prey and containing two prey, with respective relative frequencies ð1ðy=NÞÞ2 , 2ð1ðy=NÞÞðy=NÞ and ðy=NÞ2 . Now assume that the omnivore only observes its prey if there are at least two prey in its perception range. Furthermore, once a prey is observed by the omnivore, it will be killed. Thus the predator omnivore will observe a perception range ‘‘empty’’ with probability ð1ðy=NÞÞ2 , and observe a perception range with prey with probability ðy=NÞ2 (see Table 2). Here TS is the searching time, TP the handling time of pollen and TI the handling time of insect. Now the average time is Tðh,yÞ ¼ T S þ ð1hÞT P p0 þ hT I y 2 N : Using the methodology of Garay and Móri (2010), for a fixed pair (h,y), we obtain that the omnivore’s respective functional responses to pollen and prey are Pðh,yÞ : ¼ ð1hÞp0 hðy=NÞ2 and Fðh,yÞ :¼ : Tðh,yÞ Tðh,yÞ Observe that the latter function response is Holling type III. 3.2. Optimal foraging strategy Let us assume that the omnivore can optimize its nutrient intake instantaneously, in particular, during the time of finding its optimal foraging strategy h, the prey density does not change. According to Liebig’s Law, the growth rate of the omnivore is determined by the minimum nutrient intake, thus for a fixed prey density y, the fitness of the omnivore using foraging strategy h(y) as the value of h the following minimum is attained: min n1 Pðh,yÞ þ n2 Fðs,yÞ; c1 Pðh,yÞ þ c2 Fðh,yÞ : If the pollen supply is fixed, there are two possibilities. 36 J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42 3.2.1. Switching density of prey Similar to that in Section 3.1, the omnivore starts feeding on the herbivore when the herbivore density is greater than 3.2.2. Optimal mixed diet If the prey is abundant, i.e. y4y0, the minimum condition defining h(y) we obtain sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1 p0 T S y0 ¼ N : ðT S þ T P p0 Þn2 n1 p0 T I hðyÞ :¼ ðn1 c1 Þp0 2 : ðn1 c1 Þp0 þðc2 n2 Þ y=N Now, the functional responses determined by the optimal numerical responses are the following: PðhðyÞ,yÞ ¼ ðc2 n2 Þðy=NÞ2 p0 , T S ðn1 c1 Þp0 þ ðc2 n2 ÞðT S þ T P p0 Þ þ ðn1 c1 Þp0 T I ðy=NÞ2 and FðhðyÞ,yÞ ¼ ðn1 c1 Þp0 ðy=NÞ2 : T S ðn1 c1 Þp0 þ ðc2 n2 ÞðT S þT P p0 Þ þ ðn1 c1 Þp0 T I ðy=NÞ2 Moreover, now the optimal numerical response is the following: nðhðyÞ,yÞ :¼ Fig. 3. Vector field and isoclines for the mixed diet dynamics with Holling type II functional response. There is only one stable coexistence. Table 1 Activity distribution (two-patch model). Empty flower Flower with pollen TS TP þTS p2 ¼ (1 h)(1 p0) p1 ¼ (1 h)p0 No insect observed Insect observed and handled TS TI þTS 2 p3 ¼ h 1 Ny p4 ¼ h ðn1 c2 n2 c1 Þp0 ðy=NÞ2 : T S ðn1 c1 Þp0 þ ðc2 n2 ÞðT S þT P p0 Þ þðn1 c1 Þp0 T I ðy=NÞ2 3.3. Population dynamics Now the dynamics reads as x_ ¼ xðmin nðh,yÞ; cðh,yÞ k1 xÞ Fðh,yÞ x : y_ ¼ y rk2 y y For y4y0, the above dynamics reads as x_ ¼ xðnðhðyÞ,yÞk1 xÞ FðhðyÞ,yÞ x : y_ ¼ y rk2 y y y 2 N Table 2 Activity distribution. Search for pollen without finding it Search for insect without finding it Search and eat pollen Search and eat insect T1 p1 T2 p2 T3 p3 T4 p4 Example 3. Let us consider now our model with parameters n1 ¼5; c1 ¼10; n2 ¼20; c2 ¼9; p0 ¼0.9; r¼ 3; TP ¼7; TS ¼5; TI ¼10; N ¼30; k1 ¼0.0044; k2 ¼0.1 (some of them are the same as in Examples 1 and 2). The intersection of the surfaces n(h,y) and c(h,y) in Fig. 4 represents the optimal foraging strategy h(y) which is also shown in Fig. 5. Fig. 4. Total nitrogen and carbon intakes, for Holling type III functional response. According to Liebig’s Law, the realized fitness of IG predator is again determined by the minimum of both nutrient income functions, at each fixed prey density. J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42 Now for the dynamics corresponding to the optimal foraging strategy h(y), the only positive equilibrium E0 ¼(163.49, 28.68) is locally asymptotically stable. The isoclines of the system are shown in Fig. 6. Example 4. If in Example 3 we change only parameter TS ¼5 to TS ¼5/900, the new model have 3 positive equilibria E1 ¼ (75.33, 0.27); E2 ¼(274.05, 4.12); and E3 ¼(277.15, 25.77); and it can be proved that E1 and E3 are locally asymptotically stable; E2 is unstable. The isoclines and the equilibria are shown in Fig. 7. Now, with the parameters of Example 4, we show in Fig. 8 how the equilibrium coexistence depends on which species arrives first. In case (a), the omnivore is released with population size 50 only when the prey from its initial value 2 already has reached a size 20, which implies coexistence at equilibrium E3. In case (b), the omnivore is released first with population size 50, reaching 70 by the time t¼0 when the prey arrives with the same initial value 2, as in case (a). In the latter case the omnivore is capable to control the prey and keep at a low level corresponding to equilibrium E1. This result also shows how an omnivore can be applied as agent in biological pest control. 37 4. Summary of our results on bistable coexistence The basic motivation of this paper was to shed a light on bistable coexistence of omnivore and its prey observed in greenhouses. First we emphasize that our model does not depend on environmental factors, displays no regime shift (Scheffer and Carpenter, 2003; Móré et al., 2009). It is well-known that the existence of alternative stable states depends on the type of functional response. In general, only Holling III guarantees bistability for the single-species dynamics (May, 1977; Ludwig et al., 1978); the novelty of present work is that we combine optimal foraging theory with stoichiometric and nutritional ecology (Raubenheimer et al., 2009) for the case when the foods are imperfectly substitutable. We would like to emphasize that our model can be considered as a generalization of the standard optimal foraging model, since in the case of perfectly substitutable resources we get back that. Furthermore, in our model setup we also use a new general method for the derivation of functional responses (Garay and Móri, 2010), which makes it possible to take account of the details of the consumption process, incorporating in this way the foraging strategy into the functional response. Fig. 5. Optimal foraging strategy h(y). Here again IG predator consumes at a rate depending on the density of its prey. Fig. 6. Vector field and isoclines for the mixed diet dynamics with Holling type III functional response, with parameters of Example 3. Here again there is a unique stable coexistence. 38 J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42 Fig. 7. Vector fields and isoclines for the mixed diet dynamics with Holling type III functional response, with parameters of Example 4. (a) General picture; (b) locally asymptotically stable equilibrium E3; (c) locally asymptotically stable equilibrium E1; (d) unstable equilibrium E2. In this case there are two stable coexistence states. We found that while Holling type II response always implies only one equilibrium, with Holling type III (when at low densities of prey, consumption grows at increasing rate) the systems may also display two equilibria, with high and low prey densities. It is also observed that the reached equilibrium depends on which species arrives first. 1. If omnivore predators arrive first, they soon reach a low equilibrium density, since certain nutrients are scarce in pollen. When the prey arrive, the fitness of the omnivore starts to increase rapidly, and once its density achieved a certain level, it would establish the prey density at a low level. This effect is very similar to apparent competition (cf. Holt, 1977; Shakya et al., 2009) in the sense that the plant produces pollen feeding the omnivore, which in its turn reduces the density of the pest damaging the plant. However, in our model two resources together dramatically increase the fitness of the omnivore predator, since the two resources are imperfectly substitutable. 2. If prey arrive first and reach a high density, then this kind of apparent competition-like effect does not work, since at high prey density the Holling II and III functional responses are close to the saturation of omnivore. Summing up: when Liebig’s Law determines the mixed diet– fitness mapping, for bistability two things are needed: first, the two resources are imperfectly substitutable, second, at low prey density the functional response to prey is convex, so it dramatically increases with prey density. This result raises the following open question: in the case of Holling II functional response, does there exist a mixed diet– fitness mapping other then that determined by Liebig’s Law, which can guarantee that the numerical response increases dramatically at low density of prey? (For e.g. a mixed diet may dramatically increase the absorption of the limiting nutrient.) We think, if there exists such mixed diet–fitness mapping, then bistable coexistence is also possible. One of our basic aims was to demonstrate when bistable coexistence is possible. Our model dynamics is in fact much more complicated than the results presented above. But in general we claim that for our model, in mathematical terms, using the index theory of planar vector fields, the following can be proved: for J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42 39 Fig. 8. The equilibrium coexistence depends on which species arrives first. (a) Pest arrives first, x0 ¼50, y0 ¼10, the system tends to E3; (b) omnivore arrives first, x0 ¼ 70, y0 ¼ 2, the system tends to E1. Now our model describes the phenomenon observed in biological control, which was the experimental starting point of our model building. bistable coexistence the existence of at least three interior equilibria is necessary. Here bistable coexistence means that either there are two locally asymptotically stable equilibria, or there are two locally asymptotically stable limit cycles. We mention that van Rijn et al. (2002), in a different setup, found that herbivore number depends on the initial numbers of predator and prey. Moreover, since the alternative stable states depend on the arrival sequence of IG predator and prey, our model can describe a minimal succession process (Beisner et al., 2003). We feel that, if a system of several interacting species having nutritional constraints are considered, in which there are alternative stable states, and the attraction basins of these locally stable states are near the origin, then according to the arrival sequence of species (assembly history), the system can reach different stable states, thus the nutritional constraints may have an effect on the community assembly, as well. 5. Discussion In this section, first we compare our model to existing ones. Then we relate our results to those on the coexistence in IGP systems. Finally, we mention some possible applications and generalizations. 5.1. Comparison of existing models There are theoretical papers in which the quality of food is also taken into account (Abrams, 1987a, b, c, d; Fox and Vasseur, 2008; van Langevelde et al., 2008; Vasseur and Fox, 2011). There are the following differences between them and the present paper. First, we consider multidimensional nutrient intake. In the literature there are theoretical works, in which it is assumed that only one nutrient limits the fitness of each consumer (Loladze et al., 2004). Second, we build up an optimal foraging model based on numerical response of the omnivore determined by the mixeddiet–fitness mapping (described by Liebig’s Law). Third, we consider imperfectly substitutable resources, thus it is not necessary to consume different foods at a given ratio, as in the case of perfectly complementary resources (cf. Abrams, 1987a, b, c, d; Fox and Vasseur, 2008; Vasseur and Fox, 2011). Furthermore, in the present approach, in the first step we calculate function responses at fix prey density, for all possible foraging strategies. These interrelated functional responses determine the two essential nutrient incomes for all possible foraging strategies. As a consequence of Liebig’s Law, we claim that optimal foraging strategy provides the same fitness from different nutrients. Abrams (1987d) also considered our biological case, but he, with a phenomenological approach, concentrated on the optimal consumption rates. Our model can be considered as a mechanism based variant of his model, since we focused on the foraging strategy which determines the consumption rates. 5.2. About coexistence in IGP systems Our result may contribute to the solution of the problem of coexistence of IG predator, IG prey and their common resource (Holt and Huxel, 2007; Křivan 2000), although we have assumed that the plant density does not change. IGP food web, on the one hand, is a competitive system, since omnivore and its prey feed on the same plant; on the other hand, it is like an apparent competitive system, as well, since both plant and herbivore are consumed by the same omnivore. If we assume that the competition for the plant is very weak, then we get a system near to ours. In this sense our case will be a particular (or ‘‘degenerate’’) case of IGP, namely when a plant provides food to an IG predator. We emphasize that IG predator also feeds at more than one trophic level, see e.g. Arim and Marquet (2004), Coll and Guershon (2002), Diehl and Feissel (2001), Holt and Polis (1997), Jordan and Scheuring (2004) and Wäckers et al. (2005). IGP also gained major interest in biological control, not only for conservation purpose but also for pest control in open-field and greenhouse crops (see e.g. the book by Brodeur and Boivin (2006) on this issue). Since the resource in a traditional IGP model never goes extinct, the question arises: why do we get stable coexistence of prey and omnivore predator in the greenhouse (with Holling II and III as well), when the plant is not a limiting factor? More precisely: why does the rare prey not die out in our system in the light of the results on IGP coexistence? In our model we use indirectly the following assumptions which can guarantee the coexistence in traditional IGP models: 1. ‘‘IG prey should be superior in competition.’’ In general, if at least one of the nutrients is scarce in plant, then IG predator population is established at a low level without its prey, thus IG predator 40 J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42 population has a weak effect on the plant biomass, implying that the competition for plant is low when the prey is rare. 2. ‘‘IG predator should gain significant fitness benefits from consumption of the IG prey.’’ In our model, IG prey contains such a mineral which is not enough in pollen. Thus, the nutrient constraints may help to solve the theoretical problems of existence of three species IG systems. This opinion is in harmony with the fact that stable coexistence of IG predator and its prey becomes possible if both are limited by the nutrient content of the resource (Diehl, 2003). We emphasize that in our model the above two assumptions of traditional IGP model are consequences of the Libieg’s Law and of the assumption that recourses are imperfectly substitutable. Furthermore, there are optimal foraging models to study the coexistence in IGP systems (Abrams and Fung, 2010; Křivan and Diehel, 2005). The basic idea is that the optimal foraging strategy (e.g. switching) may guarantee coexistence (Abrams, 2006; Carnicer et al., 2008). Krivan and Diehel (2005) used a classical optimal foraging model based on ‘‘diet rule’’: IG predator feeds on the less profitable resources only if the more profitable one is too rare. They pointed out that ‘‘y3-species coexistence requires that the intermediate consumer is a more profitable prey than the basal resource’’. In spite of the fact that in our case ‘‘less and more profitable resources’’ make no sense (each of them is scarce in one of the nutrients), the less abundant resource is always more profitable. Abrams and Fung (2010) considered another optimization model of a 3-species IGP system, in which functional responses were of Holling type II, the resources are perfectly complementary and the optimal foraging strategy of IG predator is imperfect (it does not reach its extrema, 0 and 1). They found that in this model the coexistence of IGP system is more likely. Based on the above well-known results on IGP, in our case when IG predator is intentional forager (strictly follows its optimal foraging strategy), there exist prey densities below which IG predator does not consume its prey (see Sections 2.3.1 and 3.2.1). In the case, if the switching density can guarantee the maintenance of prey population (it is needed that the plant biomass is high enough for the prey below its switching density), then the intentional foraging can stabilize the IGP system (cf. Abrams and Fung, 2010; Krivan and Diehel, 2005). However, when the IG predator is opportunistic (consumes occasionally found rare prey while looking for plant food, see Cressman and Garay (2010) and Garay and Móri (2010), the coexistence of the IGP system seems a harder question. We suspect that when the level of opportunism is not too high, then the coexistence remains possible (cf. Abrams and Fung, 2010). There is another related open question, whether imperfectly substitutable resources alone can stabilize the IGP system when predator is not an optimal forager (cf. HilleRisLambers et al., 2006; Huisman and Weissing, 1999, 2001). However, in all these three questions the appropriate dynamics should take into account the change in plant biomass, as well. In summary: in the light of the present paper, the coexistence of IGP system is more likely, when the IG predator is the optimal forager with stochiometric constraints, and IG prey and its plant are imperfectly (or perfectly) substitutable resources. Finally, we mentioned that there are two differences between our results and those of HilleRisLambers et al. (2006), who propose a few mechanisms for coexistence of omnivore and its prey, and one of them is the case when prey is an essential food source for the omnivore. We consider imperfectly substitutable resources and our omnivore is an optimal forager. 5.3. About possible applications The developed model may have an important application in the real case of the biological control in greenhouses, which is done fundamentally with omnivorous species, as it was mentioned earlier; it is also based on the fact that the majority of the species display a type-III functional response (Cabello et al., 2007, 2011; Garcia-Martin et al., 2008; Tellez et al., 2009). Our result may be used not only in biological control in greenhouse, but may be also applied in the case when plants provide food items for natural enemies of herbivores (Wäckers et al., 2005; Young et al., 1997). In this kind of mutualism it is an important point for the plant that the cost of defense is less than its benefit (Szilágyi et al., 2009). Here, the cost for the plant is its biomass consumed by the omnivore (with or without the presence of the herbivore), and its benefit is the biomass the herbivore would have consumed in absence of the predator. In our present model this is what happens. In the light of our results, we can formulate the following hypothesis: the food provided by the plant for its mutual omnivore is poor at least in one of the essential nutrients, and the pest of the plant is rich in this nutrient. This hypothesis could be chemically tested. We already found some facts in the literature which are in harmony with this hypothesis (Matras et al., 1998; Muszynska, 2001; Wäckers et al., 2008; Wäckers, 2005; Sidhu and Patton, 1970). From theoretical point of view, for the study of the evolution of the origin of plant which provided food, the tools evolutionary game theory will be needed (cf. Cressman et al., 2004; Garay and Varga, 2000; Urbani and Ramos-Jiliberto, 2010). 5.4. Possible generalizations Not only omnivores, but also many carnivores require a range of different prey to obtain a full complement of essential amino acids. For instance, wolf spider tends to prey on three species in proportions of the essential amino acids they provide in the diet (Greenstone, 1979), and the protein and lipid ratio is also important (Jensen et al., 2011). Moreover, herbivores also balance nutrient intake (Behmer, 2009; Felton et al., 2009). For instance, moose eats energy reach terrestrial vegetation, which is rather low in sodium, thus moose generally needs to consume sodium rich aquatic plants (Belovsky, 1978). Based on that, we can introduce ‘‘stochiophagous’’ animals consuming imperfectly substitutable and/or perfectly complementary resources, thus their mixed diet is determined by the nutrients absorbable from the resources and by the stochiometric constraints of ‘‘stochiophagous’’ animal. For ‘‘stochiophagous’’ animals, we think that a generalization of Abrams’s and our models can connect in a synergic way optimal foraging theory, nutritional ecology and population dynamics (Raubenheimer et al., 2007). For instance, in this paper we defined the numerical response based on Liebig’s Law. In general, the physiology of the consumer determines the mixed diet–fitness mapping. The lipid, protein and carbohydrate ratios are also important, see e.g. Raubenheimer et al. (2007); associative effects occur when one diet item affects the digestion of another diet item, see e.g. Bjorndal (1991); the body–mass ratio of the omnivore and its prey is also important, see e.g. Otto et al. (2007); toxin dilution has an important rule in diet mixing, see e.g. Hirakawa (1995) and Singer et al. (2002); dilution of allelochemicals contained in the food plants plays an important role, see e.g. Hagele and Rowle-Rahier (1999). This mapping is not necessarily as described by Liebig’s Law (cf. Logan et al., 2004; Loladze et al., 2004). If we have a mathematically treatable mixed diet–fitness function based on biological facts (e.g. on feeding experiment in labs), then our model can be generalized in the future. In this generalized model, nutrition physiology can be connected to population dynamics and optimal foraging theory, as well. The predation process is a quite complex phenomenon (cf. Abrams and Ginzburg, 2000), depending at least on functional responses (on the details of the predation actions) and on numerical responses (on energy and nutrient intake according to the diet choice of the predators). These functions yield different J. Garay et al. / Journal of Theoretical Biology 310 (2012) 31–42 population dynamics. In this paper we have introduced a general model building method which can deal with the general predation phenomenon in the case when the population change is slow in comparison with the behavior change of predator. 41 difference is more important, in our new derivation method, although the i-th activity probability (pi) is fixed, it is not specified during the derivation. This gives a freedom for us to take account of the biological details of the considered situation, as we have done in the main text. Acknowledgments References The authors acknowledge Prof. A.P. Abrams and the anonymous referees for their helpful suggestions to improve the paper. The research has been supported by the Hungarian Scientific Research Fund OTKA (K81279) and the Scientific and Technological Innovation Fund of Hungary (ES-17/2008), the Ministry of Education and Sciences of Spain (HH2008-0023). Appendix. General derivation of functional response In order to build up a population dynamics, we have to take account of the functional response and the foraging strategy of the omnivore. Now, from Garay and Móri (2010) we recall a new method of derivation of functional responses. The basic assumption of this method is that during a given short time period the density of prey does not change, thus the omnivore activity distribution is constant during this time period. Let us consider now the following probability distribution of activities of the omnivore (see Table 1). Suppose an omnivore is staying in a perception range and looking for pollen. Then it either finds pollen or it does not, which occur with probabilities p3 and p1, respectively. Denote by T1 the average time of searching, if there is no pollen in the perception range, and let T3 be the average time of searching and handling pollen, if there is any. Similarly, if the omnivore staying in a perception range is out for prey, then it will find a prey with probability p4, and it will not with probability p2. Now T2 is the average time of searching if there is no prey in the perception range, and T4 the average time of searching and handling prey, if there is. Clearly, Ti 40 for all i¼1, 2, 3, 4 will be assumed. Then, from Garay and Móri (2010) we know that, for the fixed activity distribution of omnivore p ¼(p1,p2,p3,p4), in terms of the average time dedicated to the considered activities EðTÞ :¼ P4 i ¼ 1 pi T i , the functional response of the omnivore to the pollen (i.e. the pollen intake by a unit quantity of omnivore) is PðpÞ :¼ p3 : EðTÞ Similarly, for the prey intake of the omnivore, corresponding to its activity distribution p, is FðpÞ :¼ p4 : EðTÞ Clearly, E(T)40. Now the question arises, in which way pi depends on the optimal foraging strategy of omnivore, as well as on the density of prey and pollen supply. In the forthcoming sections we will use this general form of functional responses under different conditions. At first glance, the reader may think that this method is not quite new, since renewal theory gives the same result: functional response is the amount of food items consumed on average, divided by the average duration time (see e.g. Houston and McNamara, 1999 p. 56, and the references therein). However, there are two main differences: the first one is rather technical; we used Wald’s equality, which is a description of ‘‘vertical’’ in time (considering a given time period in which the density change of prey can be neglected); unlike renewal theory, which is a description of ‘‘horizontal’’ in time (assuming that duration time tends to infinity, and the prey density does not change, since it renews). The second Abrams, P.A., 1987a. The nonlinearity of competitive effects in models of competition for essential resources. Theor. Popul. Biol. 32, 50–65. Abrams, P.A., 1987b. Alternative models of character displacement and niche shift. I. 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