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Proceedings of the 12th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE 2012 July, 2-5, 2012.

A metapopulation model of competion type

Davide Belocchio1, Giacomo Gimmelli1, Alessandro Marchino1 and Ezio

Venturino1

1 Dipartimento di Matematica “Giuseppe Peano”, Universit`a di Torino, via Carlo Alberto 10, 10123 Torino, Italy

emails: davide.belocchio@alice.it, giacomo.gimmelli@virgilio.it, marchino.alessandro@gmail.com, ezio.venturino@unito.it1

Abstract

In this paper we present and analyse a simple two populations model for migrations among two di erent environments. The interactions among populations are of competing type for resources. Further, an external agent acts on the populations by maintaining their levels at constant value. Equilibria are investigated. A su cient condition for the coexistence equilibrium is provided.

Key words: populations, competition, migrations

MSC 2000: AMS codes (92D25, 92D40)

1Model formulation

We consider two environments among which two competing populations can migrate, denoted by P and Q. Let Pi, Qi, i = 1, 2 be their sizes in the two environments. Here the subscripts denote the environments in which they live. Let each population thrive in each environment according to logistic growth, with possibly di ering reproduction rates, respectively ri for Pi and si for Qi, and carrying capacities, respectively again Ki for Pi and Hi for Qi. We take them to be di erent, since they may be influenced by the environment. Further let ai denote the interspecific competition rate for Pi due to the presence of the popolation Qi and bi denote conversely the interspecific competition rate for Qi due to the presence of the popolation Pi.

1This paper was completed and written during a visit of the fourth author at the Max Planck Institut f¨ur Physik Komplexer Systeme in Dresden, Germany. The author expresses his thanks for the facilities provided.

A metacompetition model

Let mij the migration rate from environment j to environment i for the popolation Pj and let nij be the migration rate from j to i for the Qj ’s.

The resulting model has the following form:

At this point we make a strong assumption. We suppose that there is an external agent that keeps the populations in check, by removing individuals of the two populations at rates u for the P ’s and v for the Q’s. These control activities are performed in the same way in both environments. Thus (1) gets modified as follows:

Note that in fact the removal functions u(t) and v are functions of time through all the population sizes Pi and Qi. These controls are unknown, but we can get by, since we know their aim, which is to keep both populations at the constant fixed levels P and Q. From

0 = A + B − u, 0 = C + D − v.

Substituting back into the system (2) and eliminating the variables P2 and Q2, we find

Davide Belocchio, Giacomo Gimmelli, Alessandro Marchino, Ezio Venturino

2Equilibria

Let us consider the system (3). For ease of computation, we shall make the following re-parametrizations:

Seeking equilibria, it is easily seen that the origin in the P1 − Q1 phase plane is a feasible equilibrium only for special values of the total populations P and Q stemming from

which give either the condition P = Q = 0, i.e. the ecosystem disappears, or alternatively the following values for the total population values

P = (2K2m21 − K2r2) s2 − a2K2 (2H2n21 − H2s2) , (6) a2b2H2K2 − r2s2

Q = (2H2n21 − H2s2) r2 − b2H2 (2K2m21 − K2r2). a2b2H2K2 − r2s2

A metacompetition model

Observe that a nonzero value of P in this case means that the whole population P is contained in the second environment only, as P1 = 0, and similarly for Q. The Jacobian of

(3) in the neighborhood of the origin is

and by means of the Routh-Hurwitz Criterion we can state that the stability of the origin is ensured by the conditions

1

a2b2H2K2 − r2s2 (a2H2 (b2K2 (−2m12 − 2n12 + r1 + s1) + r2 (2n21 − s2)) +s2 (b2K2 (2m21 − r2) + r2 (2m12 − 2m21 + 2n12 − 2n21 − r1 + r2 − s1 + s2))) < 0

1

a2b2H2K2 − r2s2 (a2H2 (b2K2 ((2m12 − r1) (2n12 − s1) − (2m21 − r2) (2n21 − s2))

+r2 (2n21 − s2) (−2n12 + 2n21 + s1 − s2))

− (2m12 − 2m21 − r1 + r2) s2 × (b2K2 (2m21 − r2) + r2 (2n12 − 2n21 − s1 + s2))) > 0.

In Figure 1 we report a situation leading to this equilibrium.

There are then the two boundary equilibria with only one population in environment 1, P1 or Q1, with population levels that can be calculated from

1

0 = 2 (ζ + θP1)

1

0 = 2 (α + γQ1) ,

each subject to the condition that the other equation in (3) must be satisfied. Thus substituting these values into the remaining equation in (3), a relationship between the parameters is obtained, which thus leads to the equilibria

Davide Belocchio, Giacomo Gimmelli, Alessandro Marchino, Ezio Venturino

We will not expand the latter in terms of the original parameters due to the length of the resulting equations. We just note that the denominators in general do not vanish but for degenerate cases. In addition to (8), feasibility conditions are respectively

The Jacobian matrices in the neighborhood of these boundary equilibria are respectively

Again by means of the Routh-Hurwitz criterion we can obtain the stability conditions for these equilibria, that for the point (P1, 0) have the form

θ(δ + ι) > ζ(2β + κ), θ2 ζ + δθ2ι + 2βκζ2 > θ γθ2 + 2βζι + δκζ

and for the equilibrium (0, Q1) take the form

γ(δ + ι) > α(2η + ), γ2δι + αγ2κ + 2 ηα2 > γ θγ2 + 2αδη + +α ι ,

having used the fact that γ < 0 and θ < 0.

Then, there is the coexistence equilibrium. From (4) we can see that this point may be found by intersecting the two nullclines. An analysis of these curves reveals that they are hyperbolas, apart from degenerate cases. In fact, we can write them e.g. in matrix notation

from which it is immediately seen that the determinants of the matrices in general do not

thereby justifying the above claim. But in view of the fact that apart from γ < 0 and θ < 0 all the coe cients of the two hyperbolas do not have a definite sign, conditions ensuring that at least one intersection must be present are not easy to assess. But the set

A metacompetition model

Ω = {(P1, Q1): 0 P1 P 0 Q1 Q} is invariant for the flow generated by (3). This is a 2-dimensional compact invariant subset, therefore the Poincar´e-Bendixson Theorem holds and we have that the only three possible ω-limit sets for orbits having their initial conditions in Ω must be

•a critical point,

•a limit cycle,

•a finite number of critical points c1 . . . , ck and a countable number of limit orbits (heteroclinic and/or homoclinic) whose α- and ω-limit sets belong to {c1 . . . , ck}.

Furthermore, by means of the Sard-Smale Theorem we can easily infer that the subset of the parameter space in which we can have the non-coexistence equilibria must have measure zero. Thus for almost all values of the parameters we do not have non-coexistence equilibria.

Following these results, the problem of finding the coexistence equilibrium reduces to finding a Dulac function for (3). Up to now, we have the following su cient condition for the existence of such a function: if

then the function

1

D (P1, Q1) = P1Q1

is a Dulac function for the system (3). This allows us to ensure that whenever the su cient condition holds there cannot be any limit cycle, and therefore the system must either tend to a critical point or to orbits connecting the critical points. One such instance is empirically shown in Figure 2.

We have also run simulations attempting to obtain persistent oscillations around the coexistence equilibrium, but were not successful.

3Conclusion

A metapopulation model with two patches and two competing populations migrating among them has been considered, subject to the constraint that whole populations are held at a constant value by an external agent. In particularly unfavorable circumstances the system is shown to disappear. This can occur only for a special set of parameters, satisfying condition

(5) and thereby giving the total populations at specific levels, (6).

Equilibria containing only one population in just one environment are also obtained for very specific parameter values, (8) and attain the values given in (7). One population is

Davide Belocchio, Giacomo Gimmelli, Alessandro Marchino, Ezio Venturino

thus surviving in patch 1 and the other one in patch 2. This is clearly possible because in the assumptions of the model, stating that each one of them grows logistically in absence of the competing one, we are saying that su cient resources are available for them to thrive in each environment.

The existence and stability of the coexistence equilibrium has been shown numerically, as the analysis proves to be too complicated, in view of the fact that it originates from the intersection of conic sections, which can be shown to be hyperbolas, but for which the precise determination of their position in the phase plane it too complicated. Thus even to establish su cient conditions for their feasibility is not possible. We provide only (10), which is rather based on the use of the Dulac function and the Poincar´e-Bendixson theorem.

In situations as the one presented here it is important to address the question to what is the system’s outcome if some of the connecting paths between patches are cut out. Here unfortunately we are not able to state anything precisely, since all that can be established is that by annihilating the pairs m12, m21 and n12, n21, thereby stating that one of the two populations does not migrate, or even just m12 and n12 to mean that the path from patch 2 into patch 1 is blocked, simplifications in the conditions (5), (8) and the corresponding population levels (6) and (7) are obtained. Note however that not even one of the coe cients of the conic sections obtained by setting to zero the system (4) attains a definite sign, also in these special cases. Therefore in this situation not much more can be analytically stated about the consequences of breaking interpatch communications.

References

[1]J. T. Cronin, Movement and spatial population structure of a prairie planthopper, Ecology, 84 (2003) 1179–1188.

[2]I. Hanski, Single-species spatial dynamics may contribute to long-term rarity and commonness, Ecology 66 (1985) 335–343.

[3]I. Hanski, M. Gilpin (Ed.s), Metapopulation biology: ecology, genetics and evolution, London: Academic Press, 1997.

[4]I. Hanski, A. Moilanen, T. Pakkala, M. Kuussaari, Metapopulation persistence of an endangered butterfly: a test of the quantitative incidence function model, Conservation Biology 10 (1996) 578–590.

[5]G. Lei, I. Hanski, Metapopulation structure of Cotesia melitaearum, a parasitoid of the butterfly Melitaea cinxia, Oikos 78 (1997) 91–100.

[6]R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bulletin of the Entomological Society America 15 (1969) 237–240.

A metacompetition model

Parameters: r1=25,r2=200,s1=35,s2=300,K1=50,K2=250,H1=150,H2=350,a1=0.5,a2=0.4,b1=0.6,b2=0.8,m12=0.3,m21=0.5,n12=0.1,n21=0.4,P=139.156,Q=219.188

Figure 1: The origin of the P1—Q1 phase plane is asymptotically stable for the initial condition (P1, Q1) = (35, 55) and parameter values r1 = 25, r2 = 200, s1 = 35, s2 = 300, K1 = 50, K2 = 250, H1 = 150, H2 = 350, a1 = 0.5, a2 = 0.4, b1 = 0.6, b2 = 0.8, m12 = 0.3, m21 = 0.5, n12 = 0.1, n21 = 0.4 with P = 139.156 and Q = 219.188.

Davide Belocchio, Giacomo Gimmelli, Alessandro Marchino, Ezio Venturino

Figure 2: The coexistence equilibrium (P1, Q1) = (9.85, 328.4) is stable for the initial condition (P1, Q1) = (10, 80) and r1 = 6.2, r2 = 4.85, s1 = 10.5, s2 = 3.25, K1 = 150, K2 = 170, H1 = 200, H2 = 235, a1 = 0.5, a2 = 0.2, b1 = 1, b2 = 1.1, m12 = 0.2, m21 = 0.1, n12 = 0.3, n21 = 0.3 with P = 250 and Q = 350. Note that using these parameters the su cient condition (10) for the existence of a Dulac function does not hold.

A metacompetition model

[7]H. Malchow, S. Petrovskii, E. Venturino, Spatiotemporal patterns in Ecology and Epidemiology, Boca Raton: CRC, 2008.

[8]A. Moilanen, I. Hanski, Habitat destruction and competitive coexistence in a spatially realistic metapopulation model, Journal of Animal Ecology 64 (1995) 141-144.

[9]A. Moilanen, A. Smith, I. Hanski, Long-term dynamics in a metapopulation of the American pika, American Naturalist 152 (1998) 530–542.

[10]R. L., Schooley, L. C. Branch, Spatial heterogeneity in habitat quality and crossscale interactions in metapopulations, Ecosystems 10 (2007) 846-853.

[11]J. A. Wiens, Wildlife in patchy environments: metapopulations, mosaics, and management, in D. R. McCullough (Ed.) Metapopulations and Wildlife Conservation, Washington: Island Press, 53–84, 1996.

[12]J. A. Wiens, Metapopulation dynamics and landscape ecology, in I. A. Hanski, M. E. Gilpin (Ed.s) San Diego: Academic Press, 43–62, 2007.

[13]J. Wu, Modeling dynamics of patchy landscapes: linking metapopulation theory, landscape ecology and conservation biology, in Yearbook in Systems Ecology (English edition) Beijing: Chinese Academy of Sciences, 1994.