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Vlastimil Krivan
Vlastimil Krivan
Professor of applied mathematics
Prírodovedecká fakulta JU
Branisovská 1645/31a
370 05 Ceské Budejovice
Pracovna K3,205
Tel: [+420] 38 903 2223
E-mail: vlastimil.krivan@gmail.com
[18]Publications
[19]Curriculum vitae
[20]Lectures
[21]Community service
[22]Projects
[23]Students
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 topics
Evolutionary games where interaction times are strategy dependent
Classic matrix models of evolutionary game theory assume that all interactions
between strategies/phenotypes take the same amount of time. Here we are
developing a new methodology to study models where interaction times depend on
the two interacting strategies. We apply this theory to some classic evolutionary
games. E.g., the classic Hawk-Dove game predicts that when interaction cost
between two Hawks is low, the only evolutionarily stable strategy is all Hawks.
In other words, when cost of aggressiveness is low, all individulas will be
aggressive. However, when Hawk-Hawk interactions take long enough time, when
compared with duration of other interactions, aggressiveness evolves even when
the cost of fighting is low. 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.
[24]Krivan, V., Cressman, R. 2017. Interaction times change evolutionary
outcomes: Two-player matrix games. Journal of theoretical biology 416:199-2017
[25]Cressman, R., Krivan, V. 2019. Bimatrix games that include interaction times
alter the evolutionary outcome: The owner-intruder game. Journal of theoretical
biology 460:262-273
[26]Krivan, V., Galanthy, T. E., Cressman, R. 2018. Beyond replicator dynamics:
From frequency to density dependent models of evolutionary games. Journal of
theoretical biology 455:232-248 .
[27]Broom, M., Cressman, R., Krivan, V. 2019. Revisiting the "fallacy of
averages" in ecology: Expected gain per unit time equals expected gain divided by
expected time. Journal of Theoretical Biology 483:109993.
[28]Cressman, R., Krivan, V. (2020) Reducing courtship time promotes marital
bliss: The Battle of the Sexes game revisited with costs measured as time lost.
Journal of Theoretical Biology 503:1103826.
[29]Broom, M., Krivan, V. (2020) Two-strategy games with time constraints on
regular graphs. Journal of Theoretical Biology, 506:110426.
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.
[30]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).
[31]Krivan, V. 2014. The Allee-type Ideal Free Distribution. Journal of
Mathematical Biology 69:1497-1513.
[32]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.
[33]Cressman, R., Krivan, V. 2010. The Ideal Free Distribution as an
Evolutionarily Stable State in Density-Dependent Population Games. Oikos,
119:1231-1242.
[34]Krivan, V., Cressman, R., 2009. On evolutionary stability in prey-predator
models with fast behavioral dynamics. Evolutionary Ecology Research 11:227-251.
[35]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.
[36]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.
[37]Cressman, R., Krivan, V. 2006. Migration dynamics for the Ideal Free
Distribution. American Naturalist 168:384-397.
[38]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.
[39]Krivan, V., Sirot, E. 2002. Habitat selection by two competing species in a
two-habitat environment. American Naturalist 160:214-234.
The 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).
[40]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
[41]Krivan, V. 2010. Evolutionary stability of optimal foraging: partial
preferences in the diet and patch models. Journal of theoretical Biology
267:486-494.
[42]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.
[43]Berec, M., Krivan, V., Berec, L. 2003. Are great tits (Parus major) really
optimal foragers?. Canadian Journal of Zoology 81:780-788.
[44]Krivan, V., Eisner, J. 2003. Optimal foraging and predator-prey dynamics III.
Theoretical Population Biology 63:269-279.
[45]Krivan, V. 2000. Optimal intraguild foraging and population stability.
Theoretical Population Biology 58:79-94.
[46]Krivan, V., Sikder, A. 1999. Optimal foraging and predator-prey dynamics II.
Theoretical Population Biology 55:111-126.
[47]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.
[48]Krivan, V. 2013. Behavioral refuges and predator-prey coexistence. Journal of
Theoretical Biology 339:112-121.
[49]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.
[50]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
[51]Krivan, V. 2007. The Lotka-Volterra predator-prey model with
foraging-predation risk trade-offs. American Naturalist 170: 771-782.
Interaction webs
My interest here is to better understand biodiversity in complex interaction
webs with several interaction types.
[52]Melian, C. J., Krivan, V., Altermatt, F., Stary, P. , Pellissier, L., De
Laender, F. 2015. [53]Dispersal dynamics in food swebs. American Naturalist
185:157-168
[54]Krivan, V. 2014[55]. Competition in di- and tri-trophic food web modules.
Journal of Theoretical Biology 343:127-137.
[56]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.
[57]Melian, C. J., Bascompte, J., Jordano, P., Krivan, V. 2009. Diversity in a
complex ecological network with two interaction types. Oikos 118:122-130.
[58]Revilla, T. A., Krivan, V. 2016. Pollinator foraging flexibility and the
coexistence of competing plants. Plus One 11: e0160076.
10.1371/journal.pone.0160076
[59]Revilla, T., Krivan, V. 2018. Competition, trat-mediated facilitation, and
the structure of plant-pollinator communities. Journal of theoretical biology
440:42-57
Centrum matematické biologie
[60]Ústav matematiky
[61]Prírodovedecká fakulta
[62]Jihoceská univerzita v Ceských Budejovicích
Branisovská 1760
37005 Ceské Budejovice
Webmaster: Ludek Berec
ludek.berec(at)prf.jcu.cz
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