Vlastimil Krivan   Vlastimil Křivan
   Professor of applied mathematics

       Chair of Laboratory of Theoretical Ecology
                      Institute of Entomology
                      Biology Center
                      Czech Academy of Sciences

       Chair of Department of Mathematics and Biomathematics
                      Faculty of Science
                      University of  South Bohemia

       Address:  Biology Centre CAS
                       Branisovska 31
                       370 05 Ceske Budejovice
                       Czech republic

[+420] 38 777 5365 
       E-mail: vlastimil.krivan@gmail.com 

CV Publications Lectures  Community service Projects
Students  Courses  Some links 

My research interests

My core personal research focuses on applications of mathematics to biology and ecology.  In particular, I am interested in linking animal behavior, population dynamics and evolutionary processes. My goal is to create mathematical models that allow for a better understandig of  key mechanisms  that preserve biodiversity on Earth. To  this end I combine game theoretical methods with differential equations. In particular, I employ separation of time scales by assuming that behavioral processes operate on a much faster time scale when compared to population dynamics. This approach is complementary to the adaptive dynamics approach that assumes population dynamics operate on much faster time scale than evolution of traits.  The  approach I have developed  is based on the assumption that behavioral traits, often described as an evolutionarily stable strategy of a game instantaneously track population densities. This assumption in some cases leads to a feedback map between population densities and trait values that is multi-valued. Substituting this feedback map to population dynamics in such cases results in a differential inclusion, or a multi-valued differential equation. For low dimensional systems, these resulting models are often at least partially analyzable. 

Some research projects

Evolutionary game: Evolution of cooperation in the Prisoner's dilemma
Classic matrix models of evolutionary game theory assume that all interactions between strategies/phenotypes take the same amount of time. In Krivan and Cressman (2017) we develop a new methodology to study models where interaction times depend on the two interacting strategies. We apply this theory to the Hawk-Dove and repeated Prisoner's dilemma games. For the Hawk-Dove game aggressiveness evolves even when the cost of fighting is low if interactions take long enough time, which is not possible in the classic Hawk-DOve model. For the repeated Prisoner's dilemma, cooperation evolves if individuals opt out against defectors. This means that if a cooperator meets another cooperators, they will stay together as long as possible. However, if a cooperator meets a defector, it will play the game only once and then the pair will disband.

Krivan, V., Cressman, R. 2017. Interaction times change evolutionary outcomes: Two-player matrix games. Journal of theoretical biology 416:199-2017

Lac operon as a test of the optimal foraging theory when foragers undergo population dynamics

In Krivan (2006)  I model bacterial growth on a mixture of two sugars. It is well know that in mixed substrates with glucose and lactose bacteria often  utilize glucose first and then switch to lactose (or to some alternate source of energy). At the molecular level this switch is known as the lac operon. In this article I ask: Is this switch evolutionarily optimized? In other words, do bacteria switch between the resources at the time that  maximizes their fitness? To answer this question I build a model of bacterial growth on two substrates. The model assumes adaptive bacterial switching that maximizes bacterial per capita population growth rate - a proxy for bacterial fitness. Using some data from the literature, this model allows me to predict the time at which bacteria should switch. Then I compare this predicted time with observed times of switching for different substrates and different initial sugar concentrations. The observed times of switching show a very good agreement with predicted times. This strongly supports the idea that the molecular mechanism regulating resource switching is evolutionarily optimized.  This is also a  test of an optimal foraging theory  when populations undergo population dynamics. On contrary to the majority of experiments on the optimal foraging theory that do not consider population dynamics of foragers, this model  considers all populations dynamics.

Krivan, V. 2006. The Ideal Free Distribution and bacterial growth on two substrates. Theoretical Population Biology 69:181-191. 10.1016/j.tpb.2005.07.006

The habitat selection game
The habitat selection game is a game theoretical concept that describes species distribution in22 heterogeneous environments. For a single population, Fretwell and Lucas (1970) defined the Ideal Free Distribution (IFD) in patchy environments, under which animal payoffs in all occupied patches are the same and maximal. Thus, the IFD is a Nash equilibrium of a game that we call the Habitat Selection Game. As any strategy that uses only already occupied patches will get the same fitness at the Nash Equilibrium, it is not clear if the Ideal Free Distribution is stable with respect to mutant invasions. Cressman and Krivan (2006) proved that the IFD is also an Evolutionarily Stable Strategy, i.e., resistant to mutant strategies. The habitat selection game was extended to two and multiple species. The IFD for two competing species in a two-patch environment was derived by Krivan and Sirot (2002). Cressman et al (2004) proved that this two-species IFD is also an Evolutionarily Stable Strategy for two populations. The effects of the IFD on population dynamics of two competing species was studied by Abrams et al. (2007). Evolutionarily stability under population dynamics were considered for multiple populations by Krivan and Cressman (2009) and for a single population by Cressman and Krivan (2010). Many results on habitat selection game for competing species or predator-prey interactions were reviewed in Krivan et al. (2008).

Krivan, V. 2014. The Allee-type Ideal Free Distribution. Journal of Mathematical Biology 69:1497-1513.

Cressman, R., Krivan, V. 2013. Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds. Journal of Mathematical Biology 67:329-358.

Cressman, R., Krivan, V. 2010. The Ideal Free Distribution as an Evolutionarily Stable State in Density-Dependent Population Games. Oikos, 119:1231-1242.

Krivan, V., Cressman, R., 2009. On evolutionary stability in prey-predator models with fast behavioral dynamics. Evolutionary Ecology Research 11:227-251.

Krivan,V., Cressman, R., Schneider, C. 2008. The Ideal Free Distribution: A review and synthesis of the game theoretic perspective. Theoretical Population Biology 73:403-425.

Abrams, P., Cressman, R., Krivan, V. 2007. The role of behavioral dynamics in determining the patch distributions of interacting species. American Naturalist 169:505-518.

Cressman, R., Krivan, V. 2006. Migration dynamics for the Ideal Free Distribution. American Naturalist 168:384-397.

Cressman, R., Krivan, V., Garay, J. 2004. Ideal free distributions, evolutionary games and population dynamics in multiple species environments. The American Naturalist, 164(4):473-489.

Krivan, V., Sirot, E. 2002. Habitat selection by two competing species in a two-habitat environment. American Naturalist 160:214-234.

Optimal foraging game
Optimal foraging theory (MacArthur and Pianka, 1966; Charnov, 1976; Stephens and Krebs, 1986) assumes that organisms forage in such a way as to maximize their fitness measured as energy intake rate. These models assume a homogeneous environment with several resource types that a consumer encounters sequentially, and predict the optimal consumer diet. This line of research led to the prey model (also called the ''diet choice''; Charnov, 1976). The basic assumption here is that individuals do not compete for food. The classical example of such a situation is the experiment with great tits where a single animal feeds on two food types delivered on a conveyor belt (Krebs et al., 1977; Berec et al., 2003) which assures that prey are not depleted by predation. Certainly, this is a very unrealistic assumption, and I am interested to understand how predictions of the optimal foraging theory are shaped when population dynamics of resources and/or consumers are considered (Krivan, 1996; Krivan and Sikder, 1999; Krivan and Eisner, 2003).  The game theoretical approach to optimal foraging is presented in Cressman et al. (2014).

Cressman, R., Krivan, V., Garay, J., Brown, J. 2014. Game-theoretic methods for functional response and optimal foraging behavior. PLoS ONE 9(2): e88773. doi:10.1371/journal.pone.0088773

Krivan, V. 2010. Evolutionary stability of optimal foraging: partial preferences in the diet and patch models. Journal of theoretical Biology 267:486-494.

Krivan, V., Vrkoc, I. 2004. Should handled prey be considered? Some consequences for functional response, predator-prey dynamics and optimal foraging theory. Journal of theoretical Biology, 227:167-174.

Berec, M., Krivan, V., Berec, L. 2003. Are great tits (Parus major) really optimal foragers?. Canadian Journal of Zoology 81:780-788.

Krivan, V., Eisner, J. 2003. Optimal foraging and predator-prey dynamics III. Theoretical Population Biology 63:269-279.

Krivan, V. 2000. Optimal intraguild foraging and population stability. Theoretical Population Biology 58:79-94.

Krivan, V., Sikder, A. 1999. Optimal foraging and predator-prey dynamics II. Theoretical Population Biology 55:111-126.

Krivan, V. 1996. Optimal foraging and predator-prey dynamics. Theoretical Population Biology 49:265-290.

The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs

The Lotka-Volterra predator-prey model is one of the earliest and, perhaps, the best known example used to explain why predators can indefinitely coexist with their prey. The population cycles resulting from this model are well known. In this article I show how adaptive behavior of prey and predators can destroy these cycles and stabilize population dynamics at an equilibrium. The classical predator-prey model assumes that interaction strength between prey and predators is fixed, which means that coefficients describing interactions between prey and predators do not change in time. However, there is increasing evidence that individuals adjust their activity levels in response to predation risk and availability of resources. For example, a high predation risk due to large predator numbers leads to prey behaviors that make them less vulnerable. They can either move to a refuge or become vigilant. However, such avoidance behaviors usually also decrease animal opportunities to forage which leads to foraging-predation risk trade-off. The present article shows that such a trade-off can have a strong bearing on population dynamics. In fact, while the classical Lotka-Volterra model has isoclines that are straight lines, the foraging-predation risk trade-off leads to prey (predator) isoclines with vertical (horizontal) segments. Rosenzweig and MacArthur in their seminal work on graphical stability analysis of predator-prey models showed that such isoclines have stabilizing effect on population dynamics because they limit maximum possible fluctuations in prey and predator populations. The present article shows that not only population fluctuations are limited, but they can even be completely eliminated.

Krivan, V. 2013. Behavioral refuges and predator-prey coexistence. Journal of Theoretical Biology 339:112-121.

Krivan, V. 2011. On the Gause predator-prey model with a refuge: A fresh look at the history. Journal of Theoretical Biology 274:67-73.

Krivan, V., Pryiadarshi, A. 2015. L-shaped prey isocline in the Gause predator-prey experiments with a prey refuge. Journal of theoretical biology 370:21-26

Krivan, V. 2007. The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs. American Naturalist 170: 771-782.

Food webs
My interest here is to better understand biodiversity in  complex food webs, both theoretical  as well as  empirical.

Melian, C. J., Krivan, V., Altermatt, F., Stary, P. , Pellissier, L., De Laender, F. 2015. Dispersal dynamics in food webs. American Naturalist 185:157-168

Krivan, V. 2014. Competition in di- and tri-trophic food web modules. Journal of Theoretical Biology 343:127-137.

Berec, L., Eisner, J., Krivan, V. 2010. Adaptive foraging does not always lead to more complex food webs. Journal of Theoretical Biology 266:211-218.