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. 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.
Effects of Spatial Heterogeneity and Behavioral Tactics on Dynamics of Two Consumers and One Common Resource Thuy 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 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 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. 1. Introduction 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 1 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... most 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 densityindependent 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 important 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 density 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]). 2 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... As 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 competition 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. 2. Model 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 R dR = ε rR 1 − − aC1C R − bC2C R dτ K dC1C = (mC1R − m(C2C )C1C ) + ε[−m1C C1C + aeRC1C ] dτ dC1R (2.1) = (m(C2C ) − mC1R C1C ) + ε[−m1R C1 ] dτ dC2C = (kC2R − k(C1C )C2C ) + ε[−m2C C2C + bf RC2C ] − εlC2C dτ dC2R = (k(C1C )C2C − kC2R ) − εm2R C2R dτ 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 miR are 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 3 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... i 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 m2C m1C < min K, (2.2) ae bf (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 C1 instead of C1C . The model then reads as follows: R dR = ε rR 1 − − aC1 R − bC2C R dτ K dC1 = ε[−m1 C1 + aeRC1 ] dτ (2.3) C2C = (kC2R − k(C1 )C2C ) + ε[−m2C C2C + bf RC2C ] dτ C2R = (k(C )C − kC ) − εm C 1 2C 2R 2R 2R dτ and the condition (2.2) now becomes m1 m2C < min K, ae bf 4 (2.4) Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... We 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 kC2R − (αC1 + α0 )C2C = 0 A straightforward calculation leads to the equilibrium as follows: αC1 + α0 αC1 + α0 C2R = C2 = C2 αC1 + α0 + k H(C1 ) k k C2C = C2 = C2 αC + α + k H(C ) 1 0 1 C = C2C + C2R 2 (2.5) (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 R bk =R r 1− − aC1 − C2 dt K H(C1 ) dC1 (2.7) = C1 [−m1 + aeR] dt dC C2 2 = − km2C + m2R (αC1 + α0 ) + bf kR dt H(C1 ) 2.1.3. Stability analysis (see Appendix B) From condition (2.4) it follows that m1 m2C (αC1∗ + α0 )m2R < min K, + ae bf bf k This inequality ensures that system (2.3) has only one equilibrium, (R1∗ , 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. 5 Nguyen-Phuong and Nguyen-Ngoc 2.2. Effects of spatial heterogeneity and... 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: R dR = ε rR 1 − − aRC1C − bRC2 dτ K dC1C dτ = (−m(C2 )C1C + mC1R ) + ε[−m1C C1C + RaeC1c ] (2.8) dC1R = (m(C2 )C1C − mC1R ) − εmR C1R dτ dC2 = ε[−m C + bf RC ] − εlC 2 2 2 2 dτ and the condition (2.2) now becomes m1C m2 < min K, ae bf 2.2.1. (2.9) Fast equilibrium The fast and stable equilibrium is given as follows: βC2 + β0 C1R = C1 L(C 2) m C1 C1C = L(C2 ) C1 = C1C + C1R where L(C2 ) = βC2 + β0 + m 6 (2.10) Nguyen-Phuong and Nguyen-Ngoc 2.2.2. Effects of spatial heterogeneity and... Aggregated model The reduced slow model reads as follows: dR R am =R r 1− − C1 − bC2 dt K L(C2 ) dC1 C1 = [−(m1C m + m1R β0 ) − m1R βC2 + aemR] dt L(C2 ) dC2 = C2 [−(m2 + l) + bf R] dt 2.2.3. (2.11) 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. (R2∗ , 0, C2∗ ) is the unique non-negative and stable equilibrium of the aggregated model (2.11) provided m2 + l (m1C m + m1R β0 ) + m1R βC2∗ < min K, bf aem In summary, LIE becomes GSE and provokes LSE to extinct provided m1C m2 < min K, bf ae m +l 2 bf 3. < min K, (m1C m + m1R β0 ) + aem (2.12) m1R βC2∗ Discussion and Conclusion 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 competition 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 7 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... assumed that for larger LIE density, the larger is the migration flow of the LSE from the competition 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 consumers. 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 complicated 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. References [1] P. Amarasekare. Interactions between local dynamics and dispersal: Insights from single species models. Theoret. Popul. Biol. 53 (1998), 44–59. 8 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... [2] P. Amarasekare. Spatial dynamics in a host- multi-parasitoid community. J. Animal Ecol. 69 (2000), 201–213. [3] P. Amarasekare. Coexistence of competing parasitoids on a patchily distributed host: Local versus spatial mechanisms. Ecology 81 (2000), 1286–1296. [4] P. Amarasekare and R.M. Nisbet. Spatial heterogeneity, source-sink dynamics, and the local coexistence of competing species. The American Naturalist 158 (2001), 572–584. [5] P. Amarasekare. The role of density-dependent dispersal in source-sink dynamics. J. Theoret. Biol. 226(2004), 159–168. [6] P. Amarasekare. Spatial variation and density- dependent dispersal in competitive coexistence. Proc. Roy. Soc. London, Series B 271 (2004), 1497–1506. [7] P. Auger and D. Pontier. Fast game theory coupled to slow population dynamics: the case of domestic cat populations. Math. Biosci. 148 (1998), 6582. [8] P. Auger and R. Roussarie. Complex ecological models with simple dynamics: From individuals to populations. Acta Biotheoretica 42 (1994), 111. [9] P. Auger, R. Bravo de la Parra and E. Sanchez. Behavioral dynamics of two interacting hawkdove populations. Math. Model Meth. Appl. Sci. (M3AS) 11 (2001), 645–661. [10] P. Auger, R. Bravo de la Parra, S. Morand and E. Sanchez. A predator-prey model with predators using hawk and dove tactics. Math. Biosci. 177 and 178 (2002), 185–200. [11] P. Auger, B. Kooi, R. Bravo de la Parra and J.C. Poggiale.Bifurcation analysis of a predatorprey model with predators using hawk and dove tactics. J. Theoret. Biol. 238 (2006), 597– 607. [12] P. Auger, R. Bravo de la Parra, J.C. Poggicale, E. Sanchez and T. Nguyen Huu. Aggregation of Variables and Applications to Population Dynamics. Eds. Magal, P. and Ruan, S., Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, Vol. 1936, Mathematical Biosciences Subseries (Springer, Berlin),2008, 209–263. [13] P. Auger, R. Bravo de la Parra, J.C. Poggiale, E. Sanchez and L. Sanz.Aggregation methods in dynamical systems variables and applications in population and community dynamics. Phys. Life Rev. 5 (2008), 79–105. [14] P. Auger, C. Lett, A. Moussaoui and S. Pioch. Optimal number of sites in artificial pelagic multi- site fisheries. Canadian J. Fisheries and Aquatic Sciences, 67 (2010), 296–303. [15] K. Dao Duc, P. Auger and T. Nguyen Huu. Predator density dependent prey dispersal in a patchy environment with a refuge for the prey. South African J. Sci. 104 (2008), 180–184. 9 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... [16] A. Elabdlaoui, P. Auger, R. Bravo de la Parra, B. Kooi and R.Mchich. Effects of densitydependent migrations on stability of a two-patch predator-prey model. Math. Biosci. 210 (2007), 335–354. [17] V. Grimm and S.F. Railsback. Individual-based Modeling and Ecology. Princeton University Press, 2005. [18] Y. Iwasa, V. Andreasen and S.A. Levin. Aggregation in model ecosystems I. Perfect aggregation. Ecolog. Model. 37 (1987), 287–302. [19] Y. Iwasa, V. Andreasen and S.A. Levin. Aggregation in model ecosystems II. Approximate aggregation. IMA. J. Math. Appl. Med. Biol. 6 (1989), 1–23. [20] D.F. Lott. Intraspecific Variation in the Social Systems of Wild Vertebrates. Cambridge University, New York, 1991. [21] O. Liberg. Predation and social behaviour in a population of domestic cats: an evolutionary perspective. Ph.D. thesis, University of Lund, 1981. [22] O. Liberg and M. Sandell. Spatial organization and reproductive tactics in the domestic cat and other fields. In: Turner, D.C., Bateson, P. (Eds.), The Domestic Cat. Cambridge University, 1998. [23] I.M. Jnosi and I. Scheuring. On the evolution of density dependent dispersal in a spatially structured population model. J. Theoret. Biol. 187 (1997), 397. [24] R. Mchich, P. Auger, R. Bravo de la Parra and N. Raissi. Dynamics of a fishery on two fishing zones with fish stock dependent migrations: Aggregation and control. Ecol. Model. 158 (2002), 51–62. [25] R. Mchich, P. Auger and J.C. Poggiale. Effect of predator density dependent dispersal of prey on stability of a predator-prey system. Math. Biosci. 206 (2007), 343–356. [26] T. Namba. Density-dependent dispersal and spa- tial distribution of a population. J. Theoret. Biol. 86 (1980), 351–363. [27] D. Nguyen-Ngoc, T. Nguyen-Huu and P. Auger. Effects of refuges and density dependent dispersal on interspecific competition dynamics. International Journal of Bifurcation and Chaos, 1250029, 22(2), 2012. [28] P. Perret and J. Blondel. Experimental evidence of the territorial defense hypothesis in insular blue tits. Experientia 49 (1), 1993, 94–98. [29] D. Pontier, N. Rioux and A. Heizmann. Evidence of selection on the orange allele in the domestic cat Felis catus-The role of social- structure. Oikos 73 (3), 1995, 299–308. 10 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... [30] D. Pontier, P. Auger, R. Bravo de la Parra and E. Sanchez. The impact of behavioral plasticity at individual level on domestic cat population dynamics. Ecol. Model. 133, 2000, 117–124. [31] J.A. Silva, M.L. De Castro and D.A.R. Justo. Stability in a metapopulation model with density-dependent dispersal. Bull. Math. Biol. 63 (2001), 485–505. [32] J.A. Stamps and M. Buechner. The territorial defense hypothesis and the ecology of insular vertebrates. Q. Rev. Biol. 60 (2), 1985, 155–181. [33] L.L. Wolf and E. Waltz. Alternative mating tactics in male white- faced dragonflies: Experimental evidence for a behavioural assessment ess. Anim. Behav. 46, 1993, 325–334. [34] A. Yamane, T. Doi and Y. Ono. Mating behaviors, courtship rank and mating success of male feral cat (Felis catus). J. Ethol. 14, 1996, 35–44. 11 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... Appendix A. Equilibria and local stability analysis of single-patch model Single-patch model Single-patch model is obtained from the complete model 2.1 by neglecting the dynamics on the refuges and the migration dynamics. The single-patch model is therefore written as follows: or else R R = rR 1 − − aRC1 − bRC2 dt K dC1 = −m1 C1 + aeRC1 dt dC2 = −m + bf RC 2 2 dt (A.1) R dR =R r 1− − aC1 − bC2 dt K dC1 = C1 (−m1 + aeR) dt dC2 = C (−m + bf R) 2 2 dt (A.2) Jacobian matrix 2rR r− − aC1 − bC2 −aR −bR K J(R, C1 , C2 ) = aeC1 −m1 + aeR 0 bf C2 0 −m2 + bf R Equilibria and stability • (0, 0, 0) r 0 0 0 J(0, 0, 0) = 0 −m1 0 0 −m2 The matrix has one positive eigenvalue R thus (0, 0, 0) is unstable. 12 (A.3) Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... • (K, 0, 0) −r −ak −bk 0 J(K, 0, 0) = 0 −m1 + aek 0 0 −m2 + aek The matrix has three eigenvalues: λ1 = −r < 0, λ2 = −m1 + aeK and λ3 = −m2 + bf K. Thus, (K, 0, 0) is stable if and only if the following conditions hold K < min m1 m2 , ae bf • (R1∗ , C1∗ , 0) : where R1∗ = m1 /ae, C1∗ = r/a(1 − R1∗ /K) The condition for which this equilibrium is non-negative is given by m1 /ae < K. In this case, the jacobian matrix reads as follows: R∗ −bR1∗ −r 1 −aR1∗ J(R1∗ , C1∗ , 0) = aeCK∗ 0 0 1 0 0 −m2 + bf R1∗ The matrix has one eigenvalue λ1 = −m2 + bf R1∗ and the others eigenvalues, λ2 , λ3 , are the solutions of the following equation: rR1∗ 2 λ + λ + a2 eC1∗ R1∗ = 0 K ∗ Since λ2 + λ3 = −rR1 /K < 0 and λ2 λ3 = a2 eC1∗ R1∗ > 0. Hence, these eigenvalues have negative real parts. Therefore, (R1∗ , C1∗ , 0) is stable provided −m2 + bf R1∗ < 0 ⇔ −m2 + bf m1 /ae < 0 ⇔ m1 /ae < m2 /bf . To summarize, (R1∗ , C1∗ , 0) is non-negative and stable provided m1 m2 < min K, ae bf • (R2 , 0, C2∗ ) : where R2∗ = m2 /bf, C2∗ = r/b(1 − R2∗ /K) The condition for which this equilibrium is non-negative is given by m2 /bf < K. Similarly, this equilibrium is nonnegative and is stable provided m2 m1 < min K, bf ae To summarize, species 1 wins if one of the following condition hold: m1 m2 < min K, ae bf , species 2 wins if and only if m2 m1 < min K, , bf ae and two species get extinct, i.e. (K, 0, 0) is stable, if and only if K < min m1 m2 , ae bf 13 . Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... B. Equilibria and local stability analysis of Model 1 Aggregated model dR R bk =R r 1− − aC1 − C2 dt K H(C1 ) dC1 = C1 [−m1 + aeR] dt C2 dC 2 = − km2C + m2R (αC1 + α0 ) + bf kR dt H(C1 ) dR = Rx (R, C1 , C2 ) dt dC 1 = C1 y (R) ⇔ dt C2 dC 2 = z (R, C1 ) dt H(C1 ) (B.1) (B.2) Jacobian matrix x + RxR RxC1 C1 yR y J(R, C1 , C2 ) = C2 zR z H(C1 ) − HC1 z C2 C1 H(C1 ) H(C1 )2 RxC2 0 z H(C1 ) Equilibria and stability • (0, 0, 0): r 0 0 0 −m1 0 J(0, 0, 0) = km2C + α0 m2R 0 0 − k + α0 The matrix has one positive eigenvalue r thus (0, 0, 0) is always unstable. 14 (B.3) Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... • (K, 0, 0): −r J(0, 0, 0) = 0 0 −aK − bkK k + α0 −m1 + aeK 0 bf kK − km2C − α0 m2R k + α0 The matrix has three eigenvalues: λ1 = −r < 0, λ2 = −m1 + aeK and 0 λ3 = (bf kK − km2C − α0 m2R ) / (k + α0 ) Hence, (K, 0, 0) is stable if and only if K < min m1 km2C + α0 m2R , ae bf k • (R1∗ , C1∗ , 0) : where R1∗ = m1 /ae, C1∗ = r/a(1 − R1∗ /K) The condition for which this equilibrium is non-negative is given by m1 /ae < K. In this case, the jacobian matrix reads as follows: rR1∗ bkR1∗ ∗ − − K −aR1 H(C1∗ ) ∗ ∗ J(R1 , C1 , 0) = ∗ aeC1 0 0 ∗ ∗ bf kR1 − km2C − (αC1 + α0 )m2R 0 0 H(C1∗ ) The matrix has one eigenvalue λ1 = bf kR1∗ − km2C − (αC1∗ + α0 )m2R H(C1∗ ) and the others eigenvalues, λ2 , λ3 , are the solutions of the following equation: rR1∗ λ + a2 eC1∗ R1∗ = 0 K ∗ Since λ2 + λ3 = −rR1 /K < 0 and λ2 λ3 = a2 eC1∗ R1∗ > 0. Hence, these eigenvalues have negative real parts. Therefore, (R1∗ , C1∗ , 0) is stable provided λ2 + bf kR1∗ − km2C − (αC1∗ + α0 )m2R 0. Hence, these eigenvalues have negative real parts. Therefore, (R2∗ , 0, C2∗ ) is stable provided −m1 + aeR2∗ < 0 ⇔ (km2C + α0 m2R ) m1 < bkf ae To summarize, (R2∗ , 0, C2∗ ) is non-negative and stable provided ⇔ m1 (km2C + α0 m2R ) < min K, bkf ae C. Equilibria and local stability analysis of Model 2 Aggregated model dR R am =R r 1− − C1 − bC2 dt K L(C2 ) C1 dC1 = [−(m1C m + m1R β0 ) − m1R βC2 + aemR] dt L(C2 ) dC2 = C2 [−(m2 + l) + bf R] dt 16 (C.1) Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... dR = Rx(R, C1 , C2 ) dt dC C1 1 = y(R, C2 ) ⇔ dt L(C2 ) dC2 = C z(R) 2 dt (C.2) Jacobian matrix x + RxR RxC1 RxC2 C1 y L(C ) − yL y 2 C C 2 2 J(R, C1 , C2 ) = L(C ) yR L(C ) C1 L(C2 )2 2 2 0 z C2 zR (C.3) Equilibria and stability • (0, 0, 0): 0 0 m1C m + m1R β0 0 − 0 J(0, 0, 0) = L(0) 0 0 −(m2 + l) r The matrix has one positive eigenvalue r thus (0, 0, 0) is always unstable. • (K, 0, 0): −r KxC1 KxC2 0 J(0, 0, 0) = 0 −(m1C m + m1R β0 ) + aemK 0 0 −(m2 + l) + bf K The matrix has three eigenvalues: λ1 = −r < 0, λ2 = −(m1C m + m1R β0 ) + aemK and λ3 = −(m2 + l) + bf K. Hence, (K, 0, 0) is stable if and only if K < min m1C m + m1R β0 m2 + l , aem bf 17 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... • (R1∗ , C1∗ , 0) : where R1∗ = (m1C m + m1R β0 )/aem, C1∗ = r/a(1 − R1∗ /K) The condition for which this equilibrium is non-negative is given by (m1C m + m1R β0 )/aem < K. In this case, the jacobian matrix reads as follows: amR1∗ ∗ ∗ −R1 b −rR1 /K − L(0) ∗ ∗ ameC1 m1R βC1 J(R1∗ , C1∗ , 0) = 0 − L(0) L(0) ∗ 0 0 −(m2 + l) + bf R1 The matrix has one eigenvalue λ1 = −(m2 + l) + bf R1∗ and the others eigenvalues, λ2 , λ3 , are the solutions of the following equation: λ2 + rR1∗ a2 m2 eC1∗ R1∗ λ+ =0 K L(0)2 Since λ2 + λ3 = −rR1∗ /K < 0 and λ2 λ3 = a2 m2 eC1∗ R1∗ /L(0)2 > 0. Hence, these eigenvalues have negative real parts. Therefore, (R1∗ , C1∗ , 0) is stable provided −(m2 + l) + bf R1∗ < 0 ⇔ m2 + l (m1C m + m1R β0 ) < aem bf To summarize, (R1∗ , C1∗ , 0) is non-negative and stable provided ⇔ (m1C m + m1R β0 ) m2 + l < min K, aem bf • (R2∗ , 0, C2∗ ) : where R2∗ = (m2 + l)/bf, C2∗ = r/b(1 − R2∗ /K) The condition for which this equilibrium is non-negative is given by (m2 + l)/bf < K. In this case, the jacobian matrix reads as follows: rR2∗ amR2∗ ∗ − −R2 b − K L(C2∗ ) ∗ ∗ ∗ ∗ −(m J(R2 , 0, C2 ) = 1C m + m1R β0 ) − m1R βC2 + aemR2 0 0 ∗ L(C2 ) ∗ C2 bf 0 0 18 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and... The matrix has one eigenvalue λ1 = −(m1C m + m1R β0 ) − m1R βC2∗ + aemR2∗ L(C2∗ ) and the others eigenvalues, λ2 , λ3 , are the solutions of the following equation: rR2∗ λ + λ + b2 f C2∗ R2∗ = 0 K 2 Since λ2 + λ3 = −rR2∗ /K < 0 and λ2 λ3 = b2 f C2∗ R2∗ > 0. Hence, these eigenvalues have negative real parts. Therefore, (R2∗ , 0, C2∗ ) is stable provided −(m1C m + m1R β0 ) − m1R βC2∗ + aemR2∗ [...]...Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and [30] D Pontier, P Auger, R Bravo de la Parra and E Sanchez The impact of behavioral plasticity at individual level on domestic cat population dynamics Ecol Model 133, 2000, 117–124 [31] J.A Silva, M.L De Castro and D.A.R Justo Stability in a metapopulation model with density-dependent dispersal Bull... r/b(1 − R2∗ /K) The condition for which this equilibrium is non-negative is given by m2 /bf < K Similarly, this equilibrium is nonnegative and is stable provided m2 m1 < min K, bf ae To summarize, species 1 wins if one of the following condition hold: m1 m2 < min K, ae bf , species 2 wins if and only if m2 m1 < min K, , bf ae and two species get extinct, i.e (K, 0, 0) is stable, if and only if K < min m1... male feral cat (Felis catus) J Ethol 14, 1996, 35–44 11 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and Appendix A Equilibria and local stability analysis of single-patch model Single-patch model Single-patch model is obtained from the complete model 2.1 by neglecting the dynamics on the refuges and the migration dynamics The single-patch model is therefore written as follows: or... aemR2 0 0 ∗ L(C2 ) ∗ C2 bf 0 0 18 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and The matrix has one eigenvalue λ1 = −(m1C m + m1R β0 ) − m1R βC2∗ + aemR2∗ L(C2∗ ) and the others eigenvalues, λ2 , λ3 , are the solutions of the following equation: rR2∗ λ + λ + b2 f C2∗ R2∗ = 0 K 2 Since λ2 + λ3 = −rR2∗ /K < 0 and λ2 λ3 = b2 f C2∗ R2∗ > 0 Hence, these eigenvalues have... and stability • (0, 0, 0) r 0 0 0 J(0, 0, 0) = 0 −m1 0 0 −m2 The matrix has one positive eigenvalue R thus (0, 0, 0) is unstable 12 (A.3) Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and • (K, 0, 0) −r −ak −bk 0 J(K, 0, 0) = 0 −m1 + aek 0 0 −m2 + aek The matrix has three eigenvalues: λ1 = −r < 0, λ2 = −m1 + aeK and λ3 = −m2 + bf K Thus, (K, 0, 0) is stable if and only... −(m1C m + m1R β0 ) + aemK and λ3 = −(m2 + l) + bf K Hence, (K, 0, 0) is stable if and only if K < min m1C m + m1R β0 m2 + l , aem bf 17 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and • (R1∗ , C1∗ , 0) : where R1∗ = (m1C m + m1R β0 )/aem, C1∗ = r/a(1 − R1∗ /K) The condition for which this equilibrium is non-negative is given by (m1C m + m1R β0 )/aem < K In this case, the jacobian matrix... (αC1∗ + α0 )m2R < + ae bf bf k ∗ ∗ To summarize, (R1 , C1 , 0) is non-negative and stable provided ⇔ ⇔ m1 m2C (αC1∗ + α0 )m2R < min K, + ae bf bf k 15 Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and • (R2∗ , 0, C2∗ ) : where R2∗ = (km2C +α0 m2R )/bf k, C2∗ = rH(0)/bk(1−R2∗ /K) The condition for which this equilibrium is non-negative is given by (km2C + α0 m2R )/bf k < K In this case,... C2∗ ) is non-negative and stable provided ⇔ m1 (km2C + α0 m2R ) < min K, bkf ae C Equilibria and local stability analysis of Model 2 Aggregated model dR R am =R r 1− − C1 − bC2 dt K L(C2 ) C1 dC1 = [−(m1C m + m1R β0 ) − m1R βC2 + aemR] dt L(C2 ) dC2 = C2 [−(m2 + l) + bf R] dt 16 (C.1) Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and dR... [32] J.A Stamps and M Buechner The territorial defense hypothesis and the ecology of insular vertebrates Q Rev Biol 60 (2), 1985, 155–181 [33] L.L Wolf and E Waltz Alternative mating tactics in male white- faced dragonflies: Experimental evidence for a behavioural assessment ess Anim Behav 46, 1993, 325–334 [34] A Yamane, T Doi and Y Ono Mating behaviors, courtship rank and mating success of male feral... H(C1 ) − HC1 z C2 C1 H(C1 ) H(C1 )2 RxC2 0 z H(C1 ) Equilibria and stability • (0, 0, 0): r 0 0 0 −m1 0 J(0, 0, 0) = km2C + α0 m2R 0 0 − k + α0 The matrix has one positive eigenvalue r thus (0, 0, 0) is always unstable 14 (B.3) Nguyen-Phuong and Nguyen-Ngoc Effects of spatial heterogeneity and • (K, 0, 0): −r J(0, 0, 0) = 0 0 −aK − bkK k + α0 −m1 + ... Model 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... Effects of spatial heterogeneity and i on the resource e and f are the parameter related to consumer and consumer recruitment as a consequence of consumer -resource interaction For consumer 1,... 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,