Vlastimil Krivan | University of South Bohemia (original) (raw)
Papers by Vlastimil Krivan
PLOS ONE, 2016
We use the optimal foraging theory to study coexistence between two plant species and a generalis... more We use the optimal foraging theory to study coexistence between two plant species and a generalist pollinator. We compare conditions for plant coexistence for non-adaptive vs. adaptive pollinators that adjust their foraging strategy to maximize fitness. When pollinators have fixed preferences, we show that plant coexistence typically requires both weak competition between plants for resources (e.g., space or nutrients) and pollinator preferences that are not too biased in favour of either plant. We also show how plant coexistence is promoted by indirect facilitation via the pollinator. When pollinators are adaptive foragers, pollinator's diet maximizes pollinator's fitness measured as the per capita population growth rate. Simulations show that this has two conflicting consequences for plant coexistence. On the one hand, when competition between pollinators is weak, adaptation favours pollinator specialization on the more profitable plant which increases asymmetries in plant competition and makes their coexistence less likely. On the other hand, when competition between pollinators is strong, adaptation promotes generalism, which facilitates plant coexistence. In addition, adaptive foraging allows pollinators to survive sudden loss of the preferred plant host, thus preventing further collapse of the entire community.
J Theor Biol, 2011
This article re-analyses a prey-predator model with a refuge introduced by one of the founders of... more This article re-analyses a prey-predator model with a refuge introduced by one of the founders of population ecology Gause and his co-workers to explain discrepancies between their observations and predictions of the Lotka-Volterra prey-predator model. They replaced the linear functional response used by Lotka and Volterra by a saturating functional response with a discontinuity at a critical prey density. At concentrations below this critical density prey were effectively in a refuge while at a higher densities they were available to predators. Thus, their functional response was of the Holling type III. They analyzed this model and predicted existence of a limit cycle in predator-prey dynamics. In this article I show that their model is ill posed, because trajectories are not well defined. Using the Filippov method, I define and analyze solutions of the Gause model. I show that depending on parameter values, there are three possibilities: (1) trajectories converge to a limit cycle, as predicted by Gause, (2) trajectories converge to an equilibrium, or (3) the prey population escapes predator control and grows to infinity.
Theoretical Population Biology, Apr 1, 2006
A population dynamical model describing growth of bacteria on two substrates is analyzed. The mod... more A population dynamical model describing growth of bacteria on two substrates is analyzed. The model assumes that bacteria choose substrates in order to maximize their per capita population growth rate. For batch bacterial growth, the model predicts that as the concentration of the preferred substrate decreases there will be a time at which both substrates provide bacteria with the same fitness and both substrates will be used simultaneously thereafter. Preferences for either substrate are computed as a function of substrate concentrations. The predicted time of switching is calculated for some experimental data given in the literature and it is shown that the fit between predicted and observed values is good. For bacterial growth in the chemostat, the model predicts that at low dilution rates bacteria should feed on both substrates while at higher dilution rates bacteria should feed on the preferred substrate only. Adaptive use of substrates permits bacteria to survive in the chemostat at higher dilution rates when compared with non-adaptive bacteria.
J Optimiz Theor Appl, 1994
In this note, a perturbed control problem with state constralnts depending on a parameter u is co... more In this note, a perturbed control problem with state constralnts depending on a parameter u is considered. Assuming that, for a certain value of u, there exists a viability controller, we explicitly estimate the range of variations of u for which the same controller gives viable solutions.
Ecology, 2003
... 1992, Lefcort and Blaustein 1995, Marvier 1996, Murray et al. 1997, Mesa et al. 1998, Boots a... more ... 1992, Lefcort and Blaustein 1995, Marvier 1996, Murray et al. 1997, Mesa et al. 1998, Boots and Norman 2000, Myers et al. 2000), especially in systems where parasites are transmitted from one host species to another via predation (Lafferty 1992, Lafferty and Morris 1996). ...
Ecol Model, 1993
ABSTRACT In this paper we construct a model for the starfish, Acanthaster planci, feeding on seve... more ABSTRACT In this paper we construct a model for the starfish, Acanthaster planci, feeding on several corals. The model is based on a control system with a strategy map. This allows to model food-preferences for the starfish. We show that food-preferences alone can induce robust large-amplitude periodic fluctuations.
Journal of Mathematical Biology, 1991
A mathematical method based on the G-projection of differential inclusions is used to construct d... more A mathematical method based on the G-projection of differential inclusions is used to construct dynamical models of population biology. We suppose that the system under study, not being limited by resources, may be described by a control system
Theor Pop Biol, 1996
In this paper we consider one-predator two-prey population dynamics described by a control system... more In this paper we consider one-predator two-prey population dynamics described by a control system. We study and compare conditions for permanence of the system for three types of predator feeding behaviors: (i) specialized feeding on the more profitable prey type, (ii) generalized feeding on both prey types, and (iii) optimal foraging behavior. We show that the region of parameter space leading to permanence for optimal foraging behavior is smaller than that for specialized behavior, but larger than that for generalized behavior. This suggests that optimal foraging behavior of predators may promote coexistence in predator prey systems. We also study the effects of the above three feeding behaviors on apparent competition between the two prey types. ] 1999 Academic Press
... Volume 68, Issue 2 (April). ... Journal Information. ... Gilliam, James F. 1987. Individual B... more ... Volume 68, Issue 2 (April). ... Journal Information. ... Gilliam, James F. 1987. Individual Behavior and Population Dynamics. Ecology 68:456457. [doi:10.2307/1939283]. Articles. Individual Behavior and Population Dynamics. James F. Gilliam. See full-text article at JSTOR. Cited by. ...
Biochem Syst Ecol, 2001
Sigmoid functional responses may arise from a variety of mechanisms, one of which is switching to... more Sigmoid functional responses may arise from a variety of mechanisms, one of which is switching to alternative food sources. It has long been known that sigmoid (Holling's Type III) functional responses may stabilize an otherwise unstable equilibrium of prey and predators in Lotka-Volterra models. This poses the question of under what conditions such switching-mediated stability is likely to occur. A more complete understanding of the effect of predator switching would therefore require the analysis of one-predator/twoprey models, but these are difficult to analyze. We studied a model based on the simplifying assumption that the alternative food source has a fixed density. A well-known result from optimal foraging theory is that when prey density drops below a threshold density, optimally foraging predators will switch to alternative food, either by including the alternative food in their diet (in a fine-grained environment) or by moving to the alternative food source (in a coarse-grained environment). Analyzing the population dynamical consequences of such stepwise switches, we found that equilibria will not be stable at all. For suboptimal predators, a more gradual change will occur, resulting in stable equilibria for a limited range of alternative food types. This range is notably narrow in a fine-grained environment. Yet, even if switching to alternative food does not stabilize the equilibrium, it may prevent unbounded oscillations and thus promote persistence. These dynamics can well be understood from the occurrence of an abrupt (or at least steep) change in the prey isocline. Whereas local stability is favored only be specific types of alternative * Corresponding author. Present address: . All rights reserved.
The American Naturalist, 1997
... for all (ul, u2) and (vl, v2) between zero and one. The optimal strategy (u*, v*) is a Nash e... more ... for all (ul, u2) and (vl, v2) between zero and one. The optimal strategy (u*, v*) is a Nash equilibrium, because at a Nash equilibrium no individual can unilaterally increase its fitness by changing its strategy. An invasion-proof Nash equilibrium ...
Evolutionary Ecology Research, 2003
... co-exist even at high resource carrying capacities (right-hand panels), which makes the diamo... more ... co-exist even at high resource carrying capacities (right-hand panels), which makes the diamond food web permanent. ... Competitive co-existence caused by adaptive predators ... For higher resource carrying capacities for which inequality (7) does not hold, predators feed on both ...
Theoretical Population Biology, 1997
A host parasitoid system with overlapping generations is considered. The dynamics of the system i... more A host parasitoid system with overlapping generations is considered. The dynamics of the system is described by differential equations with a control parameter describing the behavior of the parasitoids. The control parameter models how the parasitoids split their time between searching for hosts and searching for non-host food. The choice of the control parameter is based on the assumption that each parasitoid maximizes the instantaneous growth rate of the number of copies of its genotype. It is shown that optimal individual behavior of parasitoids, with respect to time sharing between hosts and food searching, may have a stabilizing effect on the host parasitoid dynamics. ] 1997 Academic Press
The American Naturalist, Dec 1, 2007
This article studies the effects of adaptive changes in predator and/or prey activities on the Lo... more This article studies the effects of adaptive changes in predator and/or prey activities on the Lotka-Volterra predator-prey population dynamics. The model assumes the classical foraging-predation risk trade-offs: increased activity increases population growth rate, but it also increases mortality rate. The model considers three scenarios: prey only are adaptive, predators only are adaptive, and both species are adaptive. Under all these scenarios, the neutral stability of the classical Lotka-Volterra model is partially lost because the amplitude of maximum oscillation in species numbers is bounded, and the bound is independent of the initial population numbers. Moreover, if both prey and predators behave adaptively, the neutral stability can be completely lost, and a globally stable equilibrium would appear. This is because prey and/or predator switching leads to a piecewise constant prey (predator) isocline with a vertical (horizontal) part that limits the amplitude of oscillations in prey and predator numbers, exactly as suggested by Rosenzweig and MacArthur in their seminal work on graphical stability analysis of predator-prey systems. Prey and predator activities in a long-term run are calculated explicitly. This article shows that predictions based on short-term behavioral experiments may not correspond to long-term predictions when population dynamics are considered.
Evolutionary Ecology Research, 2003
Ecological studies of direct and indirect interactions in food webs usually represent systems as ... more Ecological studies of direct and indirect interactions in food webs usually represent systems as unique configurations, such as keystone predation, exploitative competition, trophic cascades or intra-guild predation. Food web dynamics are then studied using model systems that are unique to the particular configuration. In an endeavour to develop a more unified theory of food web structure and function, we explore here model systems in which a consumer species forages adaptively on two resource species along a gradient of environmental productivity and predation mortality. We explore the nature of trophic interactions under three different assumptions about what constitutes a resource and the spatial distribution of resources. We first examine a consumer (herbivore) feeding on two resources (plants) that are distributed randomly in the environment. We extend this to the case in which each plant resource occurs in a discrete patch. Finally, we examine a variant of the patch selection case in which the consumer (an omnivore) feeds within and among two trophic levels. Our modelling shows that single systems of predators, adaptive herbivores and resources can display food chain and food web topologies under different levels of productivity and predator abundance. For example, adaptive omnivory causes the exploitative competition, linear food chain and multi-trophic level omnivory to be displayed by a single system. Thus, different food web topologies, normally thought to be unique configurations in nature, can be different manifestations of the same dynamical system. This suggests that tests for top-down or bottom-up control by manipulating predator abundance or nutrient supply to resources could be confounded by topological shifts in the system itself.
Theoretical Population Biology, Apr 30, 1999
In this paper we consider one-predator two-prey population dynamics described by a control system... more In this paper we consider one-predator two-prey population dynamics described by a control system. We study and compare conditions for permanence of the system for three types of predator feeding behaviors: (i) specialized feeding on the more profitable prey type, (ii) generalized feeding on both prey types, and (iii) optimal foraging behavior. We show that the region of parameter space leading to permanence for optimal foraging behavior is smaller than that for specialized behavior, but larger than that for generalized behavior. This suggests that optimal foraging behavior of predators may promote coexistence in predator prey systems. We also study the effects of the above three feeding behaviors on apparent competition between the two prey types. ] 1999 Academic Press
J Theor Biol, 2004
Predator-prey models consider those prey that are free. They assume that once a prey is captured ... more Predator-prey models consider those prey that are free. They assume that once a prey is captured by a predator it leaves the system. A question arises whether in predator-prey population models the variable describing prey population shall consider only those prey which are free, or both free and handled prey together. In the latter case prey leave the system after they have been handled. The classical Holling type II functional response was derived with respect to free prey. In this article we derive a functional response with respect to prey density which considers also handled prey. This functional response depends on predator density, i.e., it accounts naturally for interference. We study consequences of this functional response for stability of a simple predator-prey model and for optimal foraging theory. We show that, qualitatively, the population dynamics are similar regardless of whether we consider only free or free and handled prey. However, the latter case may change predictions in some other cases. We document this for optimal foraging theory where the functional response which considers both free and handled prey leads to partial preferences which are not observed when only free prey are considered. r
PLOS ONE, 2016
We use the optimal foraging theory to study coexistence between two plant species and a generalis... more We use the optimal foraging theory to study coexistence between two plant species and a generalist pollinator. We compare conditions for plant coexistence for non-adaptive vs. adaptive pollinators that adjust their foraging strategy to maximize fitness. When pollinators have fixed preferences, we show that plant coexistence typically requires both weak competition between plants for resources (e.g., space or nutrients) and pollinator preferences that are not too biased in favour of either plant. We also show how plant coexistence is promoted by indirect facilitation via the pollinator. When pollinators are adaptive foragers, pollinator's diet maximizes pollinator's fitness measured as the per capita population growth rate. Simulations show that this has two conflicting consequences for plant coexistence. On the one hand, when competition between pollinators is weak, adaptation favours pollinator specialization on the more profitable plant which increases asymmetries in plant competition and makes their coexistence less likely. On the other hand, when competition between pollinators is strong, adaptation promotes generalism, which facilitates plant coexistence. In addition, adaptive foraging allows pollinators to survive sudden loss of the preferred plant host, thus preventing further collapse of the entire community.
J Theor Biol, 2011
This article re-analyses a prey-predator model with a refuge introduced by one of the founders of... more This article re-analyses a prey-predator model with a refuge introduced by one of the founders of population ecology Gause and his co-workers to explain discrepancies between their observations and predictions of the Lotka-Volterra prey-predator model. They replaced the linear functional response used by Lotka and Volterra by a saturating functional response with a discontinuity at a critical prey density. At concentrations below this critical density prey were effectively in a refuge while at a higher densities they were available to predators. Thus, their functional response was of the Holling type III. They analyzed this model and predicted existence of a limit cycle in predator-prey dynamics. In this article I show that their model is ill posed, because trajectories are not well defined. Using the Filippov method, I define and analyze solutions of the Gause model. I show that depending on parameter values, there are three possibilities: (1) trajectories converge to a limit cycle, as predicted by Gause, (2) trajectories converge to an equilibrium, or (3) the prey population escapes predator control and grows to infinity.
Theoretical Population Biology, Apr 1, 2006
A population dynamical model describing growth of bacteria on two substrates is analyzed. The mod... more A population dynamical model describing growth of bacteria on two substrates is analyzed. The model assumes that bacteria choose substrates in order to maximize their per capita population growth rate. For batch bacterial growth, the model predicts that as the concentration of the preferred substrate decreases there will be a time at which both substrates provide bacteria with the same fitness and both substrates will be used simultaneously thereafter. Preferences for either substrate are computed as a function of substrate concentrations. The predicted time of switching is calculated for some experimental data given in the literature and it is shown that the fit between predicted and observed values is good. For bacterial growth in the chemostat, the model predicts that at low dilution rates bacteria should feed on both substrates while at higher dilution rates bacteria should feed on the preferred substrate only. Adaptive use of substrates permits bacteria to survive in the chemostat at higher dilution rates when compared with non-adaptive bacteria.
J Optimiz Theor Appl, 1994
In this note, a perturbed control problem with state constralnts depending on a parameter u is co... more In this note, a perturbed control problem with state constralnts depending on a parameter u is considered. Assuming that, for a certain value of u, there exists a viability controller, we explicitly estimate the range of variations of u for which the same controller gives viable solutions.
Ecology, 2003
... 1992, Lefcort and Blaustein 1995, Marvier 1996, Murray et al. 1997, Mesa et al. 1998, Boots a... more ... 1992, Lefcort and Blaustein 1995, Marvier 1996, Murray et al. 1997, Mesa et al. 1998, Boots and Norman 2000, Myers et al. 2000), especially in systems where parasites are transmitted from one host species to another via predation (Lafferty 1992, Lafferty and Morris 1996). ...
Ecol Model, 1993
ABSTRACT In this paper we construct a model for the starfish, Acanthaster planci, feeding on seve... more ABSTRACT In this paper we construct a model for the starfish, Acanthaster planci, feeding on several corals. The model is based on a control system with a strategy map. This allows to model food-preferences for the starfish. We show that food-preferences alone can induce robust large-amplitude periodic fluctuations.
Journal of Mathematical Biology, 1991
A mathematical method based on the G-projection of differential inclusions is used to construct d... more A mathematical method based on the G-projection of differential inclusions is used to construct dynamical models of population biology. We suppose that the system under study, not being limited by resources, may be described by a control system
Theor Pop Biol, 1996
In this paper we consider one-predator two-prey population dynamics described by a control system... more In this paper we consider one-predator two-prey population dynamics described by a control system. We study and compare conditions for permanence of the system for three types of predator feeding behaviors: (i) specialized feeding on the more profitable prey type, (ii) generalized feeding on both prey types, and (iii) optimal foraging behavior. We show that the region of parameter space leading to permanence for optimal foraging behavior is smaller than that for specialized behavior, but larger than that for generalized behavior. This suggests that optimal foraging behavior of predators may promote coexistence in predator prey systems. We also study the effects of the above three feeding behaviors on apparent competition between the two prey types. ] 1999 Academic Press
... Volume 68, Issue 2 (April). ... Journal Information. ... Gilliam, James F. 1987. Individual B... more ... Volume 68, Issue 2 (April). ... Journal Information. ... Gilliam, James F. 1987. Individual Behavior and Population Dynamics. Ecology 68:456457. [doi:10.2307/1939283]. Articles. Individual Behavior and Population Dynamics. James F. Gilliam. See full-text article at JSTOR. Cited by. ...
Biochem Syst Ecol, 2001
Sigmoid functional responses may arise from a variety of mechanisms, one of which is switching to... more Sigmoid functional responses may arise from a variety of mechanisms, one of which is switching to alternative food sources. It has long been known that sigmoid (Holling's Type III) functional responses may stabilize an otherwise unstable equilibrium of prey and predators in Lotka-Volterra models. This poses the question of under what conditions such switching-mediated stability is likely to occur. A more complete understanding of the effect of predator switching would therefore require the analysis of one-predator/twoprey models, but these are difficult to analyze. We studied a model based on the simplifying assumption that the alternative food source has a fixed density. A well-known result from optimal foraging theory is that when prey density drops below a threshold density, optimally foraging predators will switch to alternative food, either by including the alternative food in their diet (in a fine-grained environment) or by moving to the alternative food source (in a coarse-grained environment). Analyzing the population dynamical consequences of such stepwise switches, we found that equilibria will not be stable at all. For suboptimal predators, a more gradual change will occur, resulting in stable equilibria for a limited range of alternative food types. This range is notably narrow in a fine-grained environment. Yet, even if switching to alternative food does not stabilize the equilibrium, it may prevent unbounded oscillations and thus promote persistence. These dynamics can well be understood from the occurrence of an abrupt (or at least steep) change in the prey isocline. Whereas local stability is favored only be specific types of alternative * Corresponding author. Present address: . All rights reserved.
The American Naturalist, 1997
... for all (ul, u2) and (vl, v2) between zero and one. The optimal strategy (u*, v*) is a Nash e... more ... for all (ul, u2) and (vl, v2) between zero and one. The optimal strategy (u*, v*) is a Nash equilibrium, because at a Nash equilibrium no individual can unilaterally increase its fitness by changing its strategy. An invasion-proof Nash equilibrium ...
Evolutionary Ecology Research, 2003
... co-exist even at high resource carrying capacities (right-hand panels), which makes the diamo... more ... co-exist even at high resource carrying capacities (right-hand panels), which makes the diamond food web permanent. ... Competitive co-existence caused by adaptive predators ... For higher resource carrying capacities for which inequality (7) does not hold, predators feed on both ...
Theoretical Population Biology, 1997
A host parasitoid system with overlapping generations is considered. The dynamics of the system i... more A host parasitoid system with overlapping generations is considered. The dynamics of the system is described by differential equations with a control parameter describing the behavior of the parasitoids. The control parameter models how the parasitoids split their time between searching for hosts and searching for non-host food. The choice of the control parameter is based on the assumption that each parasitoid maximizes the instantaneous growth rate of the number of copies of its genotype. It is shown that optimal individual behavior of parasitoids, with respect to time sharing between hosts and food searching, may have a stabilizing effect on the host parasitoid dynamics. ] 1997 Academic Press
The American Naturalist, Dec 1, 2007
This article studies the effects of adaptive changes in predator and/or prey activities on the Lo... more This article studies the effects of adaptive changes in predator and/or prey activities on the Lotka-Volterra predator-prey population dynamics. The model assumes the classical foraging-predation risk trade-offs: increased activity increases population growth rate, but it also increases mortality rate. The model considers three scenarios: prey only are adaptive, predators only are adaptive, and both species are adaptive. Under all these scenarios, the neutral stability of the classical Lotka-Volterra model is partially lost because the amplitude of maximum oscillation in species numbers is bounded, and the bound is independent of the initial population numbers. Moreover, if both prey and predators behave adaptively, the neutral stability can be completely lost, and a globally stable equilibrium would appear. This is because prey and/or predator switching leads to a piecewise constant prey (predator) isocline with a vertical (horizontal) part that limits the amplitude of oscillations in prey and predator numbers, exactly as suggested by Rosenzweig and MacArthur in their seminal work on graphical stability analysis of predator-prey systems. Prey and predator activities in a long-term run are calculated explicitly. This article shows that predictions based on short-term behavioral experiments may not correspond to long-term predictions when population dynamics are considered.
Evolutionary Ecology Research, 2003
Ecological studies of direct and indirect interactions in food webs usually represent systems as ... more Ecological studies of direct and indirect interactions in food webs usually represent systems as unique configurations, such as keystone predation, exploitative competition, trophic cascades or intra-guild predation. Food web dynamics are then studied using model systems that are unique to the particular configuration. In an endeavour to develop a more unified theory of food web structure and function, we explore here model systems in which a consumer species forages adaptively on two resource species along a gradient of environmental productivity and predation mortality. We explore the nature of trophic interactions under three different assumptions about what constitutes a resource and the spatial distribution of resources. We first examine a consumer (herbivore) feeding on two resources (plants) that are distributed randomly in the environment. We extend this to the case in which each plant resource occurs in a discrete patch. Finally, we examine a variant of the patch selection case in which the consumer (an omnivore) feeds within and among two trophic levels. Our modelling shows that single systems of predators, adaptive herbivores and resources can display food chain and food web topologies under different levels of productivity and predator abundance. For example, adaptive omnivory causes the exploitative competition, linear food chain and multi-trophic level omnivory to be displayed by a single system. Thus, different food web topologies, normally thought to be unique configurations in nature, can be different manifestations of the same dynamical system. This suggests that tests for top-down or bottom-up control by manipulating predator abundance or nutrient supply to resources could be confounded by topological shifts in the system itself.
Theoretical Population Biology, Apr 30, 1999
In this paper we consider one-predator two-prey population dynamics described by a control system... more In this paper we consider one-predator two-prey population dynamics described by a control system. We study and compare conditions for permanence of the system for three types of predator feeding behaviors: (i) specialized feeding on the more profitable prey type, (ii) generalized feeding on both prey types, and (iii) optimal foraging behavior. We show that the region of parameter space leading to permanence for optimal foraging behavior is smaller than that for specialized behavior, but larger than that for generalized behavior. This suggests that optimal foraging behavior of predators may promote coexistence in predator prey systems. We also study the effects of the above three feeding behaviors on apparent competition between the two prey types. ] 1999 Academic Press
J Theor Biol, 2004
Predator-prey models consider those prey that are free. They assume that once a prey is captured ... more Predator-prey models consider those prey that are free. They assume that once a prey is captured by a predator it leaves the system. A question arises whether in predator-prey population models the variable describing prey population shall consider only those prey which are free, or both free and handled prey together. In the latter case prey leave the system after they have been handled. The classical Holling type II functional response was derived with respect to free prey. In this article we derive a functional response with respect to prey density which considers also handled prey. This functional response depends on predator density, i.e., it accounts naturally for interference. We study consequences of this functional response for stability of a simple predator-prey model and for optimal foraging theory. We show that, qualitatively, the population dynamics are similar regardless of whether we consider only free or free and handled prey. However, the latter case may change predictions in some other cases. We document this for optimal foraging theory where the functional response which considers both free and handled prey leads to partial preferences which are not observed when only free prey are considered. r