. In this paper, we consider a model consists in two consumers and one common resource in a patchy environment. We assume that two consumers compete with each other for a common resource in the common patch. Individuals of both consumers can use different strategies to compete. They can be very aggressive to the other consumer individuals. They can avoid the aggressive one and leave to a refuge. We suppose that there is no food in the refuge and thus individuals cannot survive and die. This leads to the fact that individuals in the refuge have to come back to the common patch to compete for resource. We assume that for both consumers the migration is faster than the growth and mortality in the refuge and competition in the common patch. We consider the asymmetric competition: we assume that consumer 1 is locally superior resource exploiter (LSE) and consumer 2 is locally inferior resource exploiter (LIE), i.e. without migration consumer 1 will outcompete consumer 2 in the common patch. We study two cases. The first case considers LSE density dependent migration of the LIE trying to escape competition and going to its refuge when the LSE density is large. The second case considers aggressiveness of LIE leading to LIE density dependent dispersal of the LSE. We show that under some conditions, tactic 2 can allow the LIE to survive and even provoke global extinction of the LSE.
Trang 1Thuy Nguyen-Phuong and Doanh Nguyen-Ngoc
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, No 1, Dai Co Viet Street, Hanoi, Vietnam
Emails:thuy−np@yahoo.com ; doanhbondy@gmail.com
Abstract In this paper, we consider a model consists in two consumers and one common resource
in a patchy environment We assume that two consumers compete with each other for a common resource in the common patch Individuals of both consumers can use different strategies to com-pete They can be very aggressive to the other consumer individuals They can avoid the aggressive one and leave to a refuge We suppose that there is no food in the refuge and thus individuals can-not survive and die This leads to the fact that individuals in the refuge have to come back to the common patch to compete for resource We assume that for both consumers the migration is faster than the growth and mortality in the refuge and competition in the common patch We consider the asymmetric competition: we assume that consumer 1 is locally superior resource exploiter (LSE) and consumer 2 is locally inferior resource exploiter (LIE), i.e without migration consumer 1 will out-compete consumer 2 in the common patch We study two cases The first case considers LSE density dependent migration of the LIE trying to escape competition and going to its refuge when the LSE density is large The second case considers aggressiveness of LIE leading to LIE density dependent dispersal of the LSE We show that under some conditions, tactic 2 can allow the LIE
to survive and even provoke global extinction of the LSE
Key words: competition model, aggregation of variables, time scales, behavioral tactics, spatial heterogeneity
Individuals’ behaviors play an important role in population dynamics It is a fact that individuals’ behaviors have strong effects on the system that they compose [17] Understanding the effects of individual tactics that may adopt individuals at the population and community levels is one of the
Trang 2most important issues in population dynamics Individuals compete for mating, food and territory Individuals of the same population ([20], [32]) and between different populations ([32], [28], [29]) are able to use different behavioral tactics Some phenotypic characteristics, such as aggressivity, can differ between populations For instance, in urban populations, domestic cats rarely fight while
in rural populations, individuals are more likely to be aggressive for mating and to get access to some resource, ([22], [29], [7], [30], [10], [11]) Individuals, as living organisms, are capable of learning and to change tactics along their life time according to the environmental conditions, to their age, to their physical conditions and to the results of previous contests ([17], [33], [21], [34]) Behavioral plasticity allows an individual to be more flexible and to adopt the behavior that can maximize its survival in the present environmental condition
In previous works [27], we investigated the effects of aggressiveness and spatial heterogeneity
on population dynamics In this article, individuals competed for a common resource and the model was aimed at looking for the effects of behavior tactics and spatial environment (with refuges for individuals) on the outcome of the competition dynamics The model was able to show the relationship between the aggressiveness of the local inferior competitor and the case where it can invade when rare However, in this previous model, the resource was assumed to be implicit, i.e competition between species was represented by competition coefficients The aim of this work
is to take into account an explicit resource, i.e resource can be considered as a prey with logistic growth when not predated
In this contribution, we consider two consumers that exploit an explicit resource in a common patch We assume that consumer 1 is locally superior resource exploiter (LSE) Consumer 2 is therefore locally inferior resource exploiter (LIE) Hence, consumer 2 is expected to go extinct if both species would remain on this common patch all the time We are going to investigate several tactics that may be used by the LIE in order to try to avoid extinction, and globally survive:
We first study a case where LIE individuals are not aggressive so that they go to their refuge
in order to escape competition with the LSE We particularly study the case of LSE density-independent migration of LIE as well as the case of LSE density-dependent migration of LIE returning to their refuge
We also study the case where LIE individuals are aggressive with the LSE and force them to return to their refuge We thus consider the case of LIE density-dependent migration of the LSE returning to its refuge
Taking behavioral tactics by using density dependent migration into account can have impor-tant consequences on the global dynamics of the complete system For example, we refer to some earlier works in which the authors investigated these effects in the context of interaction models (both prey- predator and competition models) We refer to the article in which we considered the effects of prey density dependent migration of predators as well as predator density dependent migration of preys in a system of patches connected by fast migration events ([24], [25,], [15], [16], [27]) In these contributions, the authors assumed that preys try to avoid predators and that predators remain on the patch where prey is locally abundant These works have shown that den-sity dependent migration can have very important consequences on the global dynamics of the interaction system We also refer to some other contributions that investigated density dependent migration [1], [2], [3], [4], [5], [6], [14], [23], [26], [31])
Trang 3As in [27], we also considered two different time scales: a fast time scale corresponding to the migration between the common patch and refuge and a slow time scale corresponding to compe-tition and demography The existence of two time scales was used to reduce the dimension of the model in order to obtain an aggregated model that describes the dynamics of the total consumers densities at the slow time scale For the aggregation methods we refer to ([8], [12], [13], [18], [19])
The paper is organized as follows Section 2 presents the general model Section 2.1 studies the case of LSE density independent migration and density dependent migration of the LIE Section 2.2 focuses on the case of LIE density dependent migration of the LSE Section 3 presents a discussion
of the results and perspectives The detailed calculations of local stability analysis are given in an Appendix
We consider a model consists in two consumers and one common resource in a patchy environment
We assume that two consumers compete with each other for a common resource in the common patch Individuals of both consumers can use different strategies to compete They can be very aggressive to the other consumer individuals They can avoid the aggressive one and leave to the refuge We suppose that there is no food in the refuge and thus individuals cannot survive and die This leads to the fact that individuals in the refuge have to come back to the common patch to compete for resource We assume that for both consumers the migration is faster than the growth and mortality in the refuge and competition in the common patch In general, the dynamics of such
a model is given by
dR
dτ = ε
rR 1 − R
K − aC1CR − bC2CR
dC1C
dτ = (mC1R− m(C2C)C1C) + ε[−m1CC1C + aeRC1C]
dC1R
dτ = (m(C2C) − mC1RC1C) + ε[−m1RC1]
dC2C
dτ = (kC2R− k(C1C)C2C) + ε[−m2CC2C + bf RC2C] − εlC2C
dC2R
dτ = (k(C1C)C2C − kC2R) − εm2RC2R
(2.1)
where R is the density of the common resource r and K are the growth rate and the carrying capacity of the resource CiC and CiR are the densities of consumer i, i ∈ 1, 2, in the common patch and in the refuge, respectively miC and miRare the natural death rates of consumer i in the common patch and in the refuge The parameters a and b represent the capture rates of consumer
Trang 4i on the resource e and f are the parameter related to consumer 1 and consumer 2 recruitment as
a consequence of consumer-resource interaction For consumer 1, we suppose that m is the per capita emigration rate from the refuge to the common patch, and m(C2C) denotes the migration function from the common patch to the refuge In general, m(C2C) is assumed as an increasing function of C2C, i.e if there are too many consumer 2 in the common patch then consumer 1 is more likely to leave this patch to the refuge For consumer 2, we suppose that k is the per capita emigration rate from the refuge to the common patch, and k(C1C) denotes the migration function from the common patch to the refuge In general, k(C1C) is also assumed as an increasing function
of C1C The parameter ε represents the ratio between two time scales t = ετ In this paper, we are interested in the asymmetric competition: we assume that consumer 1 is locally superior resource exploiter (LSE) and consumer 2 is locally inferior resource exploiter (LIE), i.e without migration consumer 1 will out-compete consumer 2 in the common patch The conditions for this is given as follows
m1C
ae < min
K,m2C bf
(2.2) (see in detail in Appendix A)
2.1 Model 1: LIE individuals are not aggressive
In this model, we assume that LIE individuals are not aggressive LIE individuals play like dove while LSE individuals play like hawk (see hawk-dove game ([9], [10], [11], [30]) This leads
to the fact that LIE individuals are more likely avoid LSE individuals to go to their own refuge
We also assume that migration function of LIE is a linear function of C1C, in the other word, k(C1C) = αC1C + α0 Here, α represents the strength of density-dependence in migration, i.e if there are too many LSE individuals in the common patch then LIE individuals are more likely to leave this patch to the refuge In the case α = 0 then LIE has density-independent migration from the common patch to the refuge with the per capita emigration rate α0 For simplicity, we rewrite
C1instead of C1C The model then reads as follows:
dR
dτ = ε
rR 1 − R
K − aC1R − bC2CR
dC1
dτ = ε[−m1C1+ aeRC1]
C2C
dτ = (kC2R− k(C1)C2C) + ε[−m2CC2C + bf RC2C]
C2R
dτ = (k(C1)C2C − kC2R) − εm2RC2R
(2.3)
and the condition (2.2) now becomes
m1
ae < min
K,m2C bf
(2.4)
Trang 5We are going to use aggregation of variables methods ([12], [13]) in order to derive a reduced model The first step is to look for the existence of a stable and fast equilibrium
2.1.1 Fast equilibrium
Fast equilibrium is a solution of the following equation
A straightforward calculation leads to the equilibrium as follows:
C2R = αC1+ α0
αC1+ α0+ kC2 =
αC1+ α0 H(C1) C2
αC1+ α0+ kC2 =
k H(C1)C2
C2 = C2C + C2R
(2.6)
where H(C1) = αC1+ α0+ k
2.1.2 Aggregated model
Substitution of the fast equilibrium into the complete model (2.3) leads to a reduced model as follows:
dR
dt = R
r 1 − R
K − aC1− bk
H(C1)C2
dC1
dt = C1[−m1+ aeR]
dC2
dt =
C2 H(C1)
− km2C + m2R(αC1+ α0) + bf kR
(2.7)
2.1.3 Stability analysis (see Appendix B)
From condition (2.4) it follows that
m1
ae < min
K,m2C
bf +
(αC1∗+ α0)m2R
bf k
This inequality ensures that system (2.3) has only one equilibrium, (R∗1, C1∗, 0), which is stable
It means that LSE is always Globally Superior Resource Exploiter (GSE) Else the LSE density independent LIE migration as well as LSE density dependent LIE migration strategies are never successful in order to avoid extinction In the next subsection, we shall consider another LIE strategy, being aggressive to force LSE individuals to leave the competition patch and go to the refuge
Trang 62.2 Model 2: LIE individuals are aggressive
We are going to assume that there is a cost (l) for LIE aggressiveness which is associated with an extra-mortality for the LIE in the competition patch For simplicity, in the following model, we assume that LIE always remains on the competition patch The LSE can stay on the competition patch or come back to its refuge We also assume that migration function of LSE is a linear function of C2C, in the other word, m(C2C) = βC2C + β0 Here, β represents the strength of density-dependence in migration, i.e if there are too many LIE individuals in the common patch then LSE individuals are more likely to leave this patch to the refuge In the case β = 0 then LSE has density-independent migration from the common patch to the refuge with the per capita emigration rate β0 To avoid dealing with complex notation, we rewrite C2 instead of C2C and m2
instead of m2C The model reads as follows:
dR
dτ = ε
rR 1 − R
K − aRC1C − bRC2
dC1C
dτ = (−m(C2)C1C + mC1R) + ε[−m1CC1C + RaeC1c]
dC1R
dτ = (m(C2)C1C − mC1R) − εmRC1R
dC2
dτ = ε[−m2C2+ bf RC2] − εlC2
(2.8)
and the condition (2.2) now becomes
m1C
ae < min
K,m2 bf
(2.9)
2.2.1 Fast equilibrium
The fast and stable equilibrium is given as follows:
C1R = βC2+ β0
L(C2) C1
L(C2)C1
C1 = C1C + C1R
(2.10)
where L(C2) = βC2+ β0+ m
Trang 72.2.2 Aggregated model
The reduced slow model reads as follows:
dR
dt = R
r
1 − R K
L(C2)C1− bC2
dC1
dt =
C1 L(C2)[−(m1Cm + m1Rβ0) − m1RβC2+ aemR]
dC2
dt = C2[−(m2+ l) + bf R]
(2.11)
2.2.3 Stability analysis (see Appendix C)
We are interested in the case where LIE can survive globally when rare It is shown in Appendix C that LIE inversely becomes GSE, i.e (R∗2, 0, C2∗) is the unique non-negative and stable equilibrium
of the aggregated model (2.11) provided
m2+ l
bf < min
K,(m1Cm + m1Rβ0) + m1RβC
∗ 2
aem
In summary, LIE becomes GSE and provokes LSE to extinct provided
m1C
ae < min
K,m2 bf
m2+ l
bf < min
K,(m1Cm + m1Rβ0) + m1RβC
∗ 2
aem
(2.12)
In this contribution, we investigated different tactics that may be used by the LIE in order to avoid going extinct As a first result, model 1 demonstrated that LSE density independent migration as well as density dependent migration of LIE are not successful
The efficient tactic for the LIE is to be aggressive and to oblige the LSE to leave the compe-tition patch to its refuge In the present model, we assumed that there is a cost for the LIE to be aggressive corresponding to an extra mortality on the competition patch In this model, the LIE
is thus behaving like a hawk, aggressive all the time but paying a cost for aggressivenes On the contrary, the LSE is considered to behave like a dove, always retreating and leaving the place to the LIE but without paying any cost (see more detail about hawk-dove game theory in [9], [10], [11]) Therefore, in our model, we assumed that the LSE always returns to its refuge when the LIE
is aggressive This is a simplistic view that we investigated here as a first attempt Here, we also
Trang 8assumed that for larger LIE density, the larger is the migration flow of the LSE from the compe-tition patch to its refuge The main conclusion of this work is that this aggressive strategy of the LIE pushing away the LSE to its refuge is efficient and that under some conditions of parameters,
it allows the LIE to exclude the LSE globally
It can be seen in the condition (2.12) under which LIE becomes GSE This depends on the cost and the migration rate of LSE individuals When the cost l is small and the migration rate β is high such that the second inequality of (2.12) holds, LIE is able to provoke the LSE globally However, when the cost l is big and the migration rate β is not so high such that the inequality does not hold, LIE is not able to invade when rare
It is unlike the previous results in [27], we do not have here the coexistence of the two con-sumers The main reason is that an explicit resource was introduced in the present model This model is much more reasonable than the previous one
The present model does not take into account different tactics, hawk and dove, that may be used
by LIE as well as LSE individuals We refer to the previous contribution ([10], [11]), in which the authors studied a predator-prey model in which predators can be aggressive and dispute preys This model has shown that aggressiveness can have important consequences on the overall dynamics of the predator-prey system As a perspective, in the future, we would like to consider a new model where both LIE and LSE may use hawk and dove tactics A model of two populations using hawk and dove tactics was already studied in [9] It could be possible to couple this previous hawk-dove model and migration at a fast time scale to a classical competition model at a slow time scale
In such a competition model with hawk and dove tactics, we could consider that migration flows
of competing species could depend on the hawk density of the other species in the competition patch We also expect that the outcome of the competition globally would depend on the costs (extra mortality) of fighting for the LIE and the LSE as well as the density dependent parameters
of migration
It would be also interesting to consider several competition patches connected by migrations and not only a single competition patch as we did in this work This would lead to a more compli-cated model less tractable that would be interesting to investigate in the near future
Acknowledgements
This work was completed while the second author was staying at Vietnam Institute for Advanced Study in Mathematics (VIASM) The author would like to thank the institute for support This work was also partially supported by Vietnamese National Foundation for Science and Technology Development (NAFOSTED) under a grant
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