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Appl. Math. Inf. Sci. 9, No. 4, 1739-1752 (2015) 1739 Applied Mathematics & Information Sciences An International Journal

http://dx.doi.org/10.12785/amis/090411

On the Principle of Competitive Exclusion in Metapopulation Models

Davide Belocchio, Roberto Cavoretto, Giacomo Gimmelli, Alessandro Marchino and Ezio Venturino∗ Dipartimento di Matematica “Giuseppe Peano”, Universita di Torino, via Carlo Alberto 10, 10123 Torino, Italy

Received: 12 Oct. 2014, Revised: 12 Jan. 2015, Accepted: 13 Jan. 2015 Published online: 1 Jul. 2015

Abstract: In this paper we present and analyse a simple two populations model with migrations among two different environments. The populations interact by competing for resources. Equilibria are investigated. A proof for the boundedness of the populations is provided. A kind of competitive exclusion principle for metapopulation systems is obtained. At the same time we show that the competitive exclusion principle at the local patch level may be prevented to hold by the migration phenomenon, i.e. two competing populations may coexist, provided that only one of them is allowed to freely move or that migrations for both occur just in one direction.

Keywords: populations, , migrations, patches, competitive exclusion

1 Introduction metapopulation concept becomes essential to describe the natural interactions. This paper attempts the development In this paper we consider a minimal metapopulation of such an issue in this framework. Note that another model with two competing populations. It consists of two recent contribution in the context of patchy environments different environments among which migrations are considers also a transmissible disease affecting the allowed. populations, thereby introducing the concept of As migrations do occur indeed in nature, [6], the metaecoepidemic models, [30]. metapopulation tool has been proposed to study An interesting competition metapopulation model populations living in fragmented , [16,25]. One of with immediate patch occupancy by the strongest its most important results is the fact that a population can population and incorporating has been survive at the global level, while becoming locally proposed and investigated in [19]. Patches are created and extinct, [9,10,11,15,20,21,31,32]. An earlier, related destroyed dynamically at different rates. A completely concept, is the one of population assembly, [14], to different approach is instead taken for instance in [3], account for heterogeneous environments containing where different competition models, including distinct compositions, providing insights into facilitation, inhibition and tolerance, are investigated by issues such as and conservation. As a result, means of cellular automata. sequential slow invasion and shape successive The model we study bears close resemblance with a mixes into a persistent conguration, former model recently appeared in the literature, [23]. impenetrable by other species, [17], while, with faster However, there are two basic distinctions, in the invasions, communities change their compositions and formulation and in the analysis. As for the model each species has a chance to survive. formulation, in [23] the populations are assumed to be A specic example in nature for our competition similar species competing for an implicit . Thus situation for instance is provided by Strix occidentalis, there is a unique for both of them in which competes with, and ofter succumbs to, the larger each patch in which they reside. Furthermore their great horned owl, Bubo virginianus. The two in fact reproduction rates are the same. We remove both these compete for resources, since they share several prey, [12]. assumptions, by allowing in each patch different carrying If the environment in which they live gets fragmented, the capacities for each population, as well different competition cannot be analysed classically, and the reproduction rates. Methodologically, the approach used

∗ Corresponding author e-mail: [email protected] c 2015 NSP Natural Sciences Publishing Cor. 1740 D. Belocchio et. al.: On the Principle of Competitive Exclusion in...

in [23] uses the aggregation method, thereby reducing the 2 Model formulation system of four differential equations to a two-dimensional one, by assuming that migrations occur at a different, We consider two environments among which two faster, timescale than the life processes. This may or may competing populations can migrate, denoted by P and Q. not be the case in real life situations. In fact, referring to Let Pi, Qi, i = 1,2, their sizes in the two environments. the inhabiting the African savannas, this Here the subscripts denote the environments in which movement occurs throughout the lifetime, while they live. Let each population thrive in each environment intermingling for them does not constitute a “social” according to logistic growth, with possibly differing problem, other than the standard intraspecic competition reproduction rates, respectively ri for Pi and si for Qi, and for the resources, [26,27]. The herbivores wander in carrying capacities, respectively again Ki for Pi and Hi for search of new pastures, and the predators follow them. Qi. The latter are assumed to be different since they may This behavior might instead also be inuenced by the indeed be inuenced by the environment. Further let ai presence of predators in the surrounding areas, [29]. Thus denote the interspecic competition rate for Pi due to the the structure of African herbivores and the savanna presence of the population Qi and bi denote conversely may very well be in fact shaped by predators' the interspecic competition rate for Qi due to the behavior. presence of the population Pi. Let m the migration rate from environment j to In the current classical literature in this context, it is i j environment i for the population Pj and similarly let ni j commonly assumed that migrations of competing be the migration rate from j to i for the population Q . populations in a patchy environment lead to the situation j The resulting model has the following form: in which the superior competitor replaces the inferior one. In addition, it is allowed for an inferior competitor to P1 P1 = r1P1 1 a1P1Q1 m21P1 + m12P2 (1) invade an empty patch, but the invasion is generally − K1 − − prevented by the presence of a superior competitor in the   F1(P1,P2,Q1,Q2), patch, [28]. Based on this setting, models investigating ≡ the proportions of patches occupied by the superior and Q1 Q 1 = s1Q1 1 b1Q1P1 n21Q1 + n12Q2 inferior competitors have been set up, [13]. The effect of − H − −  1  patch removal in this context is analysed in [22], F2(P1,P2,Q1,Q2), coexistence is considered in [8,2,24,1], ≡ P2 disruptions in a realisting setting are instead studied in P2 = r2P2 1 a2P2Q2 m12P2 + m21P1 − K − − [20]. Note that in this context, the migrations are always  2  assumed to be bidirectional. Our interest here differs a bit, F3(P1,P2,Q1,Q2), ≡ since we want to consider also human artifacts or natural Q2 events that fragment the landscape, and therefore we will Q 2 = s2Q2 1 b2Q2P2 n12Q2 + n21Q1 − H − − examine particular subsystems in which migrations occur  2  F4(P1,P2,Q1,Q2). only in one direction, or are forbidden for one of the ≡ species, due to some environmental constraints. Note that a very similar model has been presented in Our analysis shows two interesting results. First of all, [23]. But (1) is more general, in that it allows different a kind of competitive exclusion principle for carrying capacities in the two patches for the two metapopulation systems also holds in suitable conditions. populations, while in [23] only one, K, is used, for both Further, the competitive exclusion principle at the local environments and populations. Further, the environments patch level may be overcome by the migration do not affect the growth rates of each individual phenomenon, i.e. two competing populations may population, while here we allow different reproduction coexist, provided that either only one of them is allowed rates for the same population in each different patch. to freely move, or that migrations for both populations Also, competition rates in [23] are the same in both occur just in one and the same direction. This shows that patches, while here they are environment-dependent. The the assumptions of the classical literature of patchy analysis technique used in [23] also makes the assumption environments may at times not hold, and this remark that there are two time scales in the model, the fast might open up new lines of investigations. dynamics being represented by migrations and the slow one by the demographics, reproduction and competition. The paper is organized as follows. In the next Section Based on this assumption, the system is reduced to a we formulate the model showing the boundedness of its planar one, by at rst calculating the equilibria of the fast trajectories. We proceed then to examine a few special part of the system using the aggregation method, and then cases, before studying the complete model: in Section 3 the aggregated two-population slow part is analysed. only one population is allowed to migrate, in Section 4 Here we thus remove the assumption of a fast the migrations occur only in one direction. Then the full migration, compared with the longer lifetime is considered in the following Section. A nal dynamics because for the large herbivores the migration discussion concludes the paper. process is a lifelong task, being always in search of new

c 2015 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 9, No. 4, 1739-1752 (2015) / www.naturalspublishing.com/Journals.asp 1741

Table 1: All the possible equilibria of the three ecosystems: Y where Pi and Qi denote the generic equilibrium point and means that the equilibrium is possible. We indicated also the unconditional instability, and with a star the instability veried 2P1 just numerically. Critical means that stability is achieved only J11 = r1 1 a1Q1 m21, − K − − under very restrictive parameter conditions, i.e. in general the  1  corresponding point must be considered unstable. P column is 2Q1 J22 = s1 1 b1P1 n21, for the model in which only P migrates, 1 2 for unidirectional − H1 − − migrations model, full for the complete→ model; unst means   2P2 unstable equilibrium, crit denotes critical equilibrium. J33 = r2 1 a2Q2 m12, − K − −  2  P1 Q1 P2 Q2 P 1 2 full → 2Q2 E1 0 0 0 0 Y (unst) Y (unst) Y (unst) J44 = s2 1 b2P2 n12. − H − − E2 0 0 + 0 – Y –  2  E3 0 0 0 + Y (unst) Y – E4 0 0 + + – Y – E5 + 0 0 0 – – – 2.1 Boundedness of the trajectories E6 + 0 + 0 Y Y Y E7 + 0 0 + – – – We will now show that the solutions of (1) are always E8 + 0 + + Y Y – E 0 + 0 0 Y (unst) – – bounded. We shall explain the proof of this assertion for 9 the complete model, but the same method can be used on E10 0 + + 0 – – – E 0 + 0 + Y Y Y each particular case, with obvious modications. 11 φ φ E12 0 + + + – Y – Let us set = P1 + Q1 + P2 + Q2. Boundedness of E13 + + 0 0 – – – implies boundedness for all the populations, since they E14 + + + 0 Y – – have to be non-negative. Adding up the system equations, E15 + + 0 + – – – we obtain a differential equation for φ, the right hand side E16 + + + + Y (unst *) Y (crit) Y (crit) of which can be bounded from above as follows

P1 Q1 φ = r1P1 1 a1P1Q1 + s1Q1 1 b1Q1P1 − K − − H −  1   1  P2 Q2 + r2P2 1 a2P2Q2 + s2Q2 1 b2P2Q2 − K2 − − H2 − pastures [7,29]. In different environments the resources     P1 Q1 are obviously different, making the statement on different r1P1 1 + s1Q1 1 ≤ − K − H carrying capacities more closely related to reality. Finally,  1   1  it is also more realistic to assume different carrying P2 Q2 + r2P2 1 + s2Q2 1 capacities for the two populations, even though they − K − H  2   2  compete for resources, as in many cases the competition r1 2 s1 2 r2 2 is only partial, in the sense that their habitats overlap, but = r1P1 P1 + s1Q1 Q1 + r2P2 P2 − K1 − H1 − K2 do not completely coincide. s2 2 + s2Q2 Q2. − H2 We will consider several subcases of this system, and (3) nally analyse it in its generality. Table 1 denes all possible equilibria of the system (1) together with the Let indication of the models in which they appear. For each ν = max r1,s1,r2,s2 , different model examined in what follows, we will { } implicitly refer to it frequently, with only changes of νK1 νH1 νK2 νH2 µ1 = , µ2 = , µ3 = , µ4 = . notation and possibly of population levels, but not for the r1 s1 r2 s2 structure of the equilibrium, i.e. the presence and absence Substituting in (3) we nd of each individual population. 2 2 2 2 φ ν ν P1 ν ν Q1 ν ν P2 ν ν Q2 For the stability analyses we will need the Jacobian of P1 µ + Q1 µ + P2 µ + Q2 µ (1), ≤ − 1 − 2 − 3 − 4 2 2 2 2 P1 Q1 P2 Q2 = ν P1 + Q1 + P2 + Q2 . − µ − µ − µ − µ  1 2 3 4 

J11 a1P1 m12 0 If we set − b1Q1 J22 0 n12 − , (2) µ3  m21 0 J33 a2P2 µ = min µi , µ+ = max µi , τ = − 0 n b Q −J − i + 4µ4  21 2 2 44  { } { } +  − 

c 2015 NSP Natural Sciences Publishing Cor. 1742 D. Belocchio et. al.: On the Principle of Competitive Exclusion in...

we nd together with s2 < 0, s1 > 0, so that also E3 is − µ3 inconditionally unstable. 2 2 2 2 φ ν P1 + Q1 + P2 + Q2 − P + Q + P + Q The point E11 = (0,H1,0,H2) is always feasible.b Two ≤ − µ4 1 1 2 2  +  eigenvalues for (2) are easily found, s1 < 0, s2 < 0. The µ3  − − 2 other ones comeb from a quadratic equation, for which the ν P1 + Q1 + P2 + Q2 − (P1 + Q1 + P2 + Q2) ≤ − 4µ4 Routh-Hurwitz conditions reduce to  +  φ r1r2 < a1H1 + a2H2 + m12 + m21, (4) = νφ 1 . − τ r1r2 + m12a1H1 + m21a2H2 + a1a2H1H2   Let us now set > r1(m12 + a2H2)+ r2(m21 + a1H1).

For parameter values satisfying these conditions then, E11 P1(0)+ Q1(0)+ P2(0)+ Q2(0)= φ(0)= u0 is stable. and let u be the solution of the Cauchy problem Equilibrium E9 = (0,H1,0,0) is always feasible, andb the Jacobian (2) has eigenvalues u(t) u(t)= νu(t) 1 , u(0)= u0. b − τ 1 √∆   λ = ( a1H1 m12 m21 + r1 + r2) , ± 2 − − − ± 2 By means of the generalized Gr¨onwall inequality we 2 φ ∆ = (a1H1 + m12 + m21 r1 r2) have that (t) u(t) for all t > 0, and so − − ≤ 4(a1H1m12 a1H1r2 m12r1 m21r2 + r1r2) limsupφ(t) limsupu(t)= τ < +∞. − − − − t +∞ ≤ t +∞ again with s1 < 0, s2 > 0 so that in view of the positivity → → − This implies at once that φ is bounded, and thus the of the last eigenvalue, E9 is always unstable. boundedness of the system's populations as desired. Existence for the equilibrium E6 can be established as Observe that the boundedness result obtained here for an intersection of curvesb in the P1 P2 phase plane. The − this minimal model is easily generalized to equations that dene them describeb the following two meta-populations living in n patches. convex parabolae

1 P1 Π1 : P2(P1) r1P1(1 ) m21P1 , ≡ m − K − 3 One population unable to migrate 12  1  1 P2 Π2 : P1(P2) r2P2(1 ) m12P2 . Here we assume that the Q population cannot migrate ≡ m21 − K2 − between the two environments. This may be due to the   fact that it is weaker, or that there are natural obstacles Both cross the coordinate axes at the origin and at another that prevent it from reaching the other environment, while point, namely these obstacles instead can be overcome by the population 1 1 P. Thus each subpopulation Q1 and Q2 is segregated in its X (r1 m21)K1,0 , W 0, (r2 m12)K2 ≡ r − ≡ r − own patch. This assumption corresponds therefore to  1   2  setting ni j = 0 into (1). In this case we will denote the respectively for Π1 and for Π2. Now by drawing these system's equilibria by Ek, with k = 1,...,16. It is easy to curves it is easily seen that they always intersect in the show that equilibria E2, E4, E10, E12 do not satisfy the rst quadrant, independently of the position of these rst equilibrium equation,b and E5, E7, E13, E15 do not points, except when both have negative coordinates. The satisfy the third one, sob thatb allb theseb points are excluded latter case need to be scrutinized more closely. To ensure from our analysis since they are unfeasible.b b b b a feasible intersection, we need to look at the parabolae At the origin, E1, the Jacobian (2) has the eigenvalues slopes at the origin. Thus, the feasible intersection exists if Π 0 Π 0 1 1 or, explicitly when √∆ 1′ ( )[ 2′ ( )]− < λ 1 b = (m12 + m21 r1 r2) , ± 2 − − ± 2 m12m21 > (m21 r1)(m12 r2). (5) 2 − − ∆ = (m12 + m21 r1 r2) 4(r1r2 m12r1 m21r2) − − − − − However, coupling this condition with the negativity of the and s > 0, s > 0, from which its instability follows. 1 2 coordinates of the above points X and W, intersections of The point E3 = (0,0,0,H2) is unconditionally feasible, the parabolae with the axes, the condition for the feasibility but the eigevalues of (2) evaluated at E3 turn out to be of E6 becomes simply b √∆ λ 1 = ( a2H2 m12 m21 + r1 + r2b) , r < m , r < m , ± 2 − − − ± 2 b 1 21 2 12 2 ∆ = (a2H2 + m12 + m21 r1 r2) − − which is exactly the assumption that the coordinates of 4(a2H2m21 a2H2r1 m12r1 m21r2 + r1r2) the points X and W be negative. Hence it is automatically − − − −

c 2015 NSP Natural Sciences Publishing Cor. Appl. Math. Inf. Sci. 9, No. 4, 1739-1752 (2015) / www.naturalspublishing.com/Journals.asp 1743

satised. Further, in the particular case in which one or axis being larger than the positive root of Π1. Letting both such points coalesce into the origin, i.e. for either 2 r1 = m21 or r2 = m12, is it easily seen that the Z = b2K2m21 (b2K2 (m12r1 + (m21 r1)r2)b corresponding parabola is tangent to the origin and a − + (r1 m21)r2s2), feasible E6 always exists. In conclusion, the equilibrium − E6 is always feasible. explicitly we have By usingb the Routh-Hurwitz criterion we can implicitlyb obtain the stability conditions as r1 > m21, a2b2H2K2 < r2s2, 2 r1s2 (b2k2 (m12 r2)+ r2s2) s2 < b2P2, s1 < b1P1, K1 < − , 2 2 Z r1 1 P1 + r2 1 P2 < m12 + m21, − K − K  1   2  2 2 together with either m12 r2 or r1 1 P1 m21 r2 1 P2 m12 > m12m21. ≥ − K1 − − K2 −        r2s2 a2H2 + m12 > r2 > m12, K2 < . Numerical simulations reveal that the stability b2 ( m12 + r2) conditions are a nonempty set, we obtain −

E6 = (119.6503,0,167.4318,0) for the parameter values 2.V > 0,L > 0,R < H2: the feasibity condition is that the r = 90.5792, r = 97.0593, s = 3.5712, s = 3.1833, 1 2 1 2 slope of Π2 at the point (0,H2) be smaller than that of K1 = 119.0779, K2 = 167.9703, H1 = 112.7548, b Π1 at the same point. But the value of the population H2 = 212.7141, a1 = 41.5414, a2 = 2.6975, P2 in thisb case would be negative, thus this condition is b1 = 39.7142, b2 = 4.1911, m12 = 0.9619, m21 = 0.9106, unfeasible;b n12 = 0, n21 = 0. 3.V > 0,L < 0,R > H2: the feasibity condition requires For the equilibrium point E8 we can dene two the slope of Π2 at the point (0,H2) to be smaller than parabolae in the P1 Q2 plane by solving the equilibrium − that of Π at the same point. But the value of the equation for P2: b 1 population Pb2 would then be negative, so that this H2b2 r1 condition is unfeasible; Π1 : Q2(P1) P1 r1 m21 P1 + H2,(6) b ≡ m12s2 − − K1 4.V > 0,L < 0,R < H2: in general there is no intersection s   point; Π : P Q 2 r s a b H K Q2 (7) b2 1( 2) 2 2 ( 2 2 2 2 2 2) 2 5.V 0 L 0 R H : the feasibity condition states that ≡ b m21K2H − < , > , > 2 2 2 Π 2 the slope of 2 at the point (0,H2) be smaller than that +(br2b2H2K2 2r2s2H2 + a2b2H K2 m12b2K2H2)Q2 2 of Π at the same point; explicitly − 2 2 − 2 1 +(r2s2H r2b2H K2 + m12b2H K2) . 2 − 2 2 b m21 > r1, r2s2 > a2b2H2K2 The rst parabola intersects the Q2 axis at the point b m12r1 (0,H2), it always has two real roots, one of which is a2H2 < + r2, a2H2 + m12 > r2. positive and the other negative, and has the vertex with m21 r1 1 1 − abscissa V = K1(R1 m21)r− . The second parabola 2 − 1 6.V < 0,L > 0,R < H2: for feasibility, the intersection intersects the Q2 axis at the points between Π2 and the P1 axis must be larger than the b2H2K2r2 H2r2s2 + b2H2K2m12 positive root of Π1; in other words R1 0, − , ≡ a2,b2H2K2 r2s2 b  −  m21 > r1, r2 > m12, a2h2 + m12 < r2, R2 (0,H2). b ≡ 2 r1s2 (b2K2 (m12 r2)+ r2s2) Given that the two parabolae always have one K1 < − , Z intersection on the boundary of the rst quadrant, we can (m21 r1)r2s2 r2s2 formulate a certain number of conditions ensuring their − < K2 < . b2 (m12r1 + (m21 r1)r2) b2 ( m12 + r2) intersection in the interior of the rst quadrant. These − − conditions arise from the abscissa of the vertex of Π1, of Π 7.V < 0,L < 0,R > H2: there can be no intersection the leading coefcient of 2 and by the relative positions point; of the roots of Π2. By denoting as mentioned by bV the 8.V < 0,L < 0,R < H2: for feasibity the slope of Π2 at abscissa of vertex of Π ,b by L the leading coefcient of 1 the point (0,H ) must be smaller than that of Π at the Π 2 1 2 and by R theb ordinate of R1, we have explicitly 8 sets same point. In this case, explicitly we haveb the of conditions: b feasibility conditions b b 1.V > 0, L > 0, R > H2: the feasibility condition reduces just to the intersection between Π2 and the P1 m21 > r1, r2 > a2H2 + m12, a2b2H2K2 > r2s2.

b c 2015 NSP Natural Sciences Publishing Cor. 1744 D. Belocchio et. al.: On the Principle of Competitive Exclusion in...

The stability conditions given by the Routh-Hurwitz and the following stability conditions given by the Routh- criterion can be stated as s1 < b1P1 together with Hurwitz criterion s2 < b2P2 together with

2r1 2P2r2 2s1 m12 + m21 + P1 b1 + + + Q1 a1 + K1 K2 H1 2P1r1 2r2     m12 + m21 + + P2 b2 + > r1 + r2 + s1, K1 K2   b1H1P1 ((K2 2P2)(K1 (r1 m21) 2P1r1)r2 2s2 − − − +Q2 a2 + > r1 + r2 + s2, K2m12 (K1 2P1)r1) H2 − −   > (H1 2Q1)(K2m12 (a1K1Q1 (K1 2P1)r1) b2H2P2 ((K2 2P2)(K1 (r1 m21) 2P1r1)r2 − − − − − − (K2 2P2)(K1 (m21 + a1Q1 r1)+ 2P1r1)r2)s1, K2m12 (K1 2P1)r1) − − − − − > (H2 2Q2)(K2 (a2K1m21Q2 (K1 2P1)(m12 − − − and nally +a2Q2)r1) (K2 2P2)(K1 (m21 r1)+ 2P1r1)r2)s2 − − − (H1 (2K2P1r1 + K1 (2P2r2 + K2 (m12 + m21 +b1P1 + a1Q1 r1 r2 s1))) − − − and nally +2K1K2Q1s1)(H1 (K2m12 (a1K1Q1 (K1 2P1)r1) − − (K2 2P2) (K1 (m21 + a1Q1 r1)+ 2P1r1)r2) − − − +b1H1P1 (2K2P1r1 + K1 (K2 (m12 + m21 r1 r2)+ 2P2r2)) (H2 (2K2P1r1 + K1 (2P2r2 + K2 (m12 + m21 + b2P2 − − (H1 2Q1)(K1 (K2 (m12 + m21 + a1Q1 r1 r2) +a2Q2 r1 r2 s2))) − − − − − − − +2K2P1r1 + 2P2r2))s1) +2K1K2Q2s2)(H2 (K2 (a2K1m21Q2 > H1K1K2 (b1H1P1 (K2m12 (2P1 K1)r1 (K1 2P1)(m12 + a2Q2)r1) − − − +(K2 2P2)( 2P1r1 + K1 ( m21 + r1))r2) (K2 2P2)(K1 (m21 r1)+ 2P1r1)r2) − − − − − − (H1 2Q1)(K2m12 (a1K1Q1 (K1 2P1)r1) +b2H2P2 (2K2P1r1 + K1 − − − − (K2 2P2)(K1 (m21 + a1Q1 r1)+ 2P1r1)r2)s1), (K2 (m12 + m21 r1 r2)+ 2P2r2)) − − − × − − with population values evaluated at equilibrium. Again, (H2 2Q2)(2K2P1r1 + K1 − − the whole set of conditions can be satised to lead to a (K2 (m12 + m21 + a2Q2 r1 r2)+ 2P2r2))s2) × − − stable conguration for the following parameter choice: > H2K1K2 (b2H2P2 ( K2m12 (K1 2P1)r1 r1 = 19.5081, r2 = 28.3773, s1 = 151.5480, − − s 164 6916, K 224 4882, K 249 8364, +(K2 2P2)( 2P1r1 + K1 ( m21 + r1))r2) 2 = . 1 = . 2 = . − − − H1 = 247.9646, H2 = 234.9984, a1 = 28.5839, (H2 2Q2)(K2 (a2K1m21Q2 − − a2 = 12.9906, b1 = 60.1982, b2 = 82.5817, (K1 2P1)(m12 + a2Q2)r1) m 0 8687, m 0 1361, n 0, n 0, with − − 12 = . 21 = . 12 = 21 = (K2 2P2)(K1 (m21 r1)+ 2P1r1)r2)s2), initial conditions (7.5967,48.9253,13.1973,16.8990). − − − The equilibrium coordinates are E14 = (0.0301,244.9973,242.1885,0),see Figure 2. where the population values are those at equilibrium. The coexistence equilibrium E16 has been deeply Also in this case the simulations show that this investigated numerically. It has been found to be always equilibrium E = (220.0633,0,0.0176,247.9334) can be feasible, but never stable for all theb sets of parameters 8 used. achieved for the parameter values r1 = 148.9386, r2 = 97.3583,b s1 = 162.3161, s2 = 94.1847, K1 = 221.5104, K2 = 260.2843, H1 = 240.0507, H2 = 252.1136, a1 = 91.3287, a2 = 49.4174, 4 Unidirectional migration only b1 = 50.0022, b2 = 88.6512, m12 = 0.0424, m21 = 0.9730, n12 = 0, n21 = 0, see Figure 1. In this case, we assume that it is not possible to migrate from patch 2 back into patch 1, so that the coefcients For the equilibrium E14 the same above analysis can m12 and n12 vanish. The reasons behind this statement can be repeated, with only changes in the parabolae and in the be found in natural situations. For instance it can be subscripts of the above explicitb feasibility conditions. The observed that freshwater shes swim downstream much details are omitted, but the results provide a set of more easily than upstream. In particular obstacles like feasibility conditions dams and waterfalls may hinder the upstream migrations. In any case the overcoming of these obstacles requires a sizeable effort, for which sufcient energy must be m12 > r2, r1 > a1H1 + m21, a1b1H1K1 > r1s1, allocated. This however may not always be available.

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We denote the equilibria here by Ek, k = 1,...,16. and with respective conditions for the non-negativity of Equilibria E5, E7, E9, E10, E13, E14, E15 are found to be their rst components given by all infeasible. e r1 m21, (13) The origine Ee1 hase twoe positivee e eigevaluese r2 > 0 and ≥ s1 n21. (14) s2 > 0, so that it is unstable. ≥ The points Ee2 = (0,0,K2,0) and E3 = (0,0,0,H2) are Note further that if (13) and (14) hold, then A,B > 0. But feasible. For the former, the eigenvalues of the Jacobian then √A > K2r1r2 and √B > H2s1s2, so that E6 and are r2, m21e+ r1, n21 + s1, b2eK2 + s2, giving the − − − − − E11 have the second component negative, i.e. they are stability conditions − infeasible. The feasibility conditions for E6+ and Ee11+ are then respectively given by (13) and (14). The eigenvalues r1 < m21, s1 < n21, s2 < b2K2. (8) e for E6+ are m21 r1 and e e − For the latter instead, the eigenvalues are m21 + r1, − b1K1 (m21 r1) √A a2H2 + r2, n21 + s1, s2, with the following n21e+ − + s1, , conditional− stability− conditions− − r1 −K2r1

b2 K2r1r2 √A r1 < m21, r2 < a2H2, s1 < n21. (9) − − + s2.  2r1r2  Equilibrium giving the stability conditions

K2s2(H2a2 r2) H2r2(b2K2 s2) r1(n21 s1) > b1K1(m21 r1), (15) E4 = 0,0, − , − − − a2b2H2K2 r2s2 a2b2H2K2 r2s2  − −  2r1r2s2 < b2 K2r1r2 + √A , is feasiblee for either one of the two alternative sets of   inequalities where we used (13). Eigenvalues of E11 are n21 s1 and + − a2H2 > r2, b2K2 > s2; (10) a1H1 (n21 s1) √B a2H2 < r2, b2K2 < s2. (11) m21 + r1 + e − , , − s1 −H2s1 The eigenvalues are m21 + r1, n21 + s1, λ , where − − ± 1 √B r + a H λ 2 2 2 2 2 s s 2(a2b2H2K2 r2s2) = r2s2 − − 1 2 ! − ± +r2s2 ( a2H2 b2K2 + s2) √∆, from which the stability conditions follow − − ± ∆ = r2s2 [r2s2 ( a2H2 b2K2 + r2 + s2)2 − − s1(m21 r1) > a1H1(n21 s1), 2r2s1s2 < a2 H2s1s2 + √B . +4(a2H2 r2)(b2K2 s2)(a2b2H2K2 r2s2)]. − − − − −  (16)  In case (10) holds, we nd λ+ > 0 so that E4 is unstable. having again used (14). In case instead of (11) the stability conditions are For the next two equilibria, we are able only to analyse e feasibility. We nd r1 < m21, s1 < n21, (12) K1m21 + K1r1 E8 = − ,0,B,A and simulations show that this point is indeed stably r1 achieved, Figure 3. for the parameter values r1 = 0.15,   r2 = 90, s1 = 0.55, s2 = 61, K1 = 250, K2 = 300, with e H = 120, H = 500, a = 12, a = 0.06, b = 3, 1 2 1 2 1 1 b2 = 0.015, m12 = 0, m21 = 0.9, n12 = 0, n21 = 0.8, A = a2H2K2r1s2 K2r1r2s2 giving the equilibrium E4 = (0,0,205,2799,474,9398). 2(a2b2H2K2r1 r1r2s2) { − 2− The next points come in pairs. They are + 4(K1K2m s2 K1K2m21r1s2)( a2b2H2K2r1+ − 21 − − e 2 1/2 K1(r1 m21) K2r1r2 √A r1r2s2) + (a2H2K2r1s2 K2r1r2s2) E6 = − ,0, ± ,0 , − ± r1 2r1r2 ! 1  o B = H2s2 e H1(s1 n21) H2s1s2 √B 2s2(a2b2H2K2r1 r1r2s2) { − E11 = 0, − ,0, ± , 2 − ± s1 2s1s2 ! a2b2H2 K2r1s2 + b2H2K2r1r2s2 2(a b H K r r r s ) wheree 2 2 2 2 1 1 2 2 −2 2 b2H2( 4(K1K2m21s2 K1K2m21r1s2)( a2b2H2K2r1 A = K2r1r2( 4K1m21 + 4K1m21r1 + K2r1r2), − − − − − 2 1/2 2 +r r s ) + (a H K r s K r r s ) , B = H2s1s2( 4H1n + 4H1n21s1 + H2s1s2), 1 2 2 2 2 2 1 2 2 1 2 2 − 21 −  o

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and 5 The complete model H1n21 + H1s1 E = 0, − ,D,C 12 s We consider now the full system (1) In this case, the points  1  where E2, E3, E4, E5, E7, E8, E9, E10, E12, E13, E14, E15 are seen e to be all infeasible. 1 At the origin E1, the characteristic polynomial factors D = b2K2H2s1r2 H2s1s2r2 2(b2a2K2H2s1 s1s2r2) { − to give the two quadratic equations − 2 2 + 4(H1H2n21r2 H1H2n21s1r2)( b2a2K2H2s1 λ λ(r1 m21 + r2 m12) + (r1 m21)(r2 m12) − − − − − − − − 2 1/2 m m = 0 +s1s2r2) + (b2K2H2s1r2 H2s1s2r2) 21 12 − − and 1  o C = K2r2 2 2r2(b2a2K2H2s1 s1s2r2) { λ λ(s1 n21 + s2 n12) + (s1 n21)(s2 n12) 2 − − − − − − b a K H s r + a K H s s r n21n12 = 0. 2 2 2 2 1 2 2 2 2 1 2 2 − − 2(b2a2K2H2s1 s1s2r2) Stability conditions are then ensured by the Routh-Hurwitz 2− a2K2( 4(H1H2n r2 H1H2n21s1r2) conditions, which explicitly become − − 21 − ( b2a2K2H2s1 + s1s2r2) m + m > r + r , r r > r m + r m , (19) ×− 21 12 1 2 1 2 1 12 2 21 2 1/2 n21 + n12 > s1 + s2, s1s2 > s1n12 + s2n21. +(b2K2H2s1r2 H2s1s2r2) . − These conditions are nevertheless incompatible, since  o 1 Feasibility for E8 is ensured by from the second one we have r1 > m21 + m12r1r2− > m21 and similarly r2 > m12, contradictingthus the rst one. The re1 > m21, A > 0, B > 0, (17) origin is therefore always unstable. The points E6 and E11 may be studied by the same while for E12 by means of (5) and therefore are always feasible. The stability of E6 is given implicitly by e s1 > n21, C > 0, D > 0. (18) s2 < b2P2, s1 < b1P1, Numerical simulations show in fact their stability, as seen 2 2 in Figures 4 and 5, respectively for the parameter values r1 1 P1 + r2 1 P2 < m12 + m21, − K1 − K2 r1 = 148.8149, r2 = 95.9844, s1 = 121.9733,     2 2 s2 = 171.8885, K1 = 228.8361, K2 = 223.9932, r1 1 P1 m21 r2 1 P2 m12 > m12m21, H = 201.4337, H = 216.7927, a = 71.2694, − K − − K − 1 2 1   1    2   a2 = 47.1088, b1 = 68.1972, b2 = 7.1445, m12 = 0, whereas for the equilibrium E11 we have the conditions m21 = 0.8175, n12 = 0, n21 = 0.5186, giving equilibrium E8 = (227.5790,0,0.0184,216.6269) and for the r2 < a2Q2, r1 < a1Q1, parameter values r1 = 70.3319, r2 = 117.0528, 2 2 s = 183.4387, s = 151.4400, K = 219.0223, s1 1 Q1 + s2 1 Q2 < n12 + n21, b1 2 1 − H1 − H2 K = 207.5854, H = 226.5399, H = 293.4011,     2 1 2 2 2 a1 = 56.8824, a2 = 1.1902, b1 = 16.2182, b2 = 31.1215, s1 1 Q1 n21 s2 1 Q2 n12 > n12n21. − H − − H − m12 = 0, m21 = 0.2630, n12 = 0, n21 = 0.4505, giving   1    2   E12 = (0,225.9835,207.5514,0.0161). Simulations were carried out to demonstrate that the The coexistence equilibrium stability conditions of these points can be satised. The Ee16 = (18.4266,18.4266,18.6164,18.6164) has been equilibrium E6 = (203.2749,0,262.4315,0) is stably numerically investigated for the parameter values achieved for the parameter values r1 = 179.2820, re1 = 100, r2 = 100, s1 = 100, s2 = 100, K1 = 250, r2 = 48.8346, s1 = 162.9841, s2 = 27.9518, K2 = 250, H1 = 250, H2 = 250, a1 = 5, a2 = 5, b1 = 5, K1 = 202.4929, K2 = 265.3457, H1 = 204.9169, b2 = 5, m12 = 0, m21 = 0.5, n12 = 0, n21 = 0.5, see Figure H2 = 203.4834, a1 = 58.2431, a2 = 69.7650, 6, from which its stability under suitable parameter values b1 = 94.4784, b2 = 77.2208, m12 = 0.9758, is shown. Note that the parameters have been chosen in a m21 = 0.5674, n12 = 0.4716, n21 = 0.2537. Equilibrium very peculiar way, the reproduction rates all coincide, as E11 = (0,245.4094,0,263.5643) is attained with the do all the carrying capacities, the competition rates and choice r1 = 46.2191, r2 = 191.5950, s1 = 70.5120, the migration rates. However, numerical experiments s2 = 171.4748, K1 = 240.3233, K2 = 256.7841, reveal that by slightly perturbing these values, the H1 = 244.9968, H2 = 263.8244, a1 = 49.1146, stability of this equilibrium point is immediately lost. We a2 = 43.3295, b1 = 77.5334, b2 = 38.0149, conclude then that the coexistence equilibrium can be m12 = 0.4620, m21 = 0.6463, n12 = 0.8896, achieved at times, but is generically unstable. n21 = 0.8370, with initial conditions

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(47.4215,86.4803,27.8785,70.8909), as can be seen in 6.2 Unrestricted migrations Figures 7 and 8. For the coexistence equilibrium Looking now more specically at each one of the proposed E16 = (10.7367,10.7367,15.0240,15.0240) we have models, we draw the following inferences. similar results as for the one of the one-migration only The model with unrestricted migration possibilities case. It is shown to exist and be stable in Figure 9, for the allows the survival of either one of the competing very specic parameter values r1 = 110, r2 = 80, populations, in both patches, E6 and E11. Coupling this s1 = 110, s2 = 80, K1 = 360, K2 = 270, H1 = 360, result with the fact that the interior coexistence has been H2 = 270, a1 = 10, a2 = 5, b1 = 10, b2 = 5, m12 = 0.5, numerically shown to be stable just for a specic m21 = 0.1, n12 = 0.5, n21 = 0.1. Its stability however is parameter choice, but it is generally unstable, this result easily broken under slight perturbations of the system appears to be an extension of the classical competitive parameters. Again, thus, the coexistence equilibrium E16 exclusion principle, [18], to metapopulation systems, in is not generically stable. agreement with the classical literature in the eld, e.g. [ 1, 2,8,13,20,22,24,28]. It is apparent here, as well as in the classical case, that an information on how the basins of 6 Conclusions attraction of the two mutually exclusive boundary equilibria is important in assessing the nal outcome of the system, based on the knowledge of its present state. 6.1 Discussion of the possible systems' To this end, relevant numerical work has been performed equilibria for two and three dimensional systems, [4,5]. An extension to higher dimensions is in progress. The metapopulation models of competition type here considered show that only a few populations congurations are possible at a stable level. First of all, i n 6.3 Migration allowed for just one population virtue of our assumptions, all these ecosystems will never disappear. Table 1 shows that equilibria E5, E7, E10, E13, For the model in which only one population can migrate, E15 cannot occur in any one of the models considered two more equilibria are possible in addition to those of here. Of these, E7 and E10 are the most interesting ones. the full model, i.e. the resident, non-migrating, population They show that one competitor cannot survive solely in Q can survive just in one patch with the migrating one, one patch, while the other one thrives alone in the second and the patch can be either one of the two in the model, patch. Thus it is not possible to reverse the outcome of a equilibria E8 and E14. The resident population cannot superior competitor in one patch in the other patch. outcompete the migrating one, since the equilibria E and Further, in the rst patch the two populations can coexist 3 E are bothb unconditionallyb unstable. Thus, when just one only in the model in which only one population is allowed 9 population migrates, the classical principle of competitive to migrate back and forth into the other patch, equilibrium b exclusion does not necessarily hold neither at the wider E . In that case, the migrating population thrives also b 14 metapopulation level, nor in one of the two patches, as alone in the second environment. The coexistence of all shown by the nonvanishing population levels of patch 2 in populations in both environments is “fragile”, it occurs only under very limited assumptions. Coexistence in the equilibrium E8 = (220.0633,0,0.0176,247.9334), Figure second patch can occur instead with the rst one empty at 1, and in patch 1 in equilibrium E4, only in the following two cases. For the E14 = (0.0301b ,244.9973,242.1885,0), Figure 2. The one-directional migration model, with immigrations into coexistence in one of the two patches appears to be the second patch, the rst patch is left empty. When the possibleb since the weaker species can migrate to the other rst patch is instead populated by one species only, at competitor-free environment, thrive there and migrate equilibria E8 for both the one-population and back to reestablish itself with the competitor in the unidirectional migrations models and at E12, again for the original environment. But the principle of competitive one-directional migrations model. The equilibria in which exclusion can in fact occur also in this model, since the one population is wiped out from the instead, numerical simulations reveal it, consider indeed the E6 and E11, occur in all three models. Finally, the three equilibrium E6 = (119.6503,0,167.4318,0). However, remaining equilibria contain only one population in just restrictions in the interpatch moving possibilities of one one patch. At E2, only for the unidirectional migration population mightb prevent its occurrence. The coexistence model, the migrating population survives in the arrival of all the populations appears to be always impossible in patch. At E9 it is the residential, i.e. the non-migrating, view of the instability of the equilibrium E16. population that survives in its own patch, only for the Using the algorithm introduced in [5], we have also one-population migrations model. At E3 for both explored a bit how the migration rates inuenceb the shape particular cases instead, the residential population of the basins of attraction of the two equilibria E6 and E11. survives in the “arrival” patch of the other migrating For this model where just one population is allowed population. to migrate, keeping the following demographicb parametersb

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200 0.04 100 0.02 Population P1 Population P1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Time 1.5 200 1 100 0.5 Population Q1 Population Q1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Time

0.04 200 0.03 100 0.02

Population P2 0.01 Population P2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Time

15 200 10 100 5 Population Q2 Population Q2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Time

Fig. 1: The two top populations P1 and Q1 occupy the rst patch, Fig. 2: The two top populations P1 and Q1 occupy the rst the bottom ones P2 and Q2 the second patch. The equilibrium patch, the bottom ones P2 and Q2 the second patch. The E8 =(220.0633,0,0.0176,247.9334) is stable for the parameter equilibrium E14 =(0.0301,244.9973,242.1885,0) is stable for values r1 = 148.9386, r2 = 97.3583, s1 = 162.3161, s2 = the parameter values r1 = 19.5081, r2 = 28.3773, s1 = 151.5480, 94b .1847, K1 = 221.5104, K2 = 260.2843, H1 = 240.0507, H2 = s2 = 164.6916,b K1 = 224.4882, K2 = 249.8364, H1 = 247.9646, 252.1136, a1 = 91.3287, a2 = 49.4174, b1 = 50.0022, b2 = H2 = 234.9984, a1 = 28.5839, a2 = 12.9906, b1 = 60.1982, 88.6512, m12 = 0.0424, m21 = 0.9730, n12 = 0, n21 = 0. b2 = 82.5817, m12 = 0.8687, m21 = 0.1361, n12 = 0, n21 = 0.

xed, 100 r1 = 6.4, s1 = 5.0, k1 = 8.0, h1 = 5.7, (20) 50 Population P1 0 0 0.05 0.1 0.15 0.2 0.25 0.3 a = 2.9, b = 2.5, r = 5.5, s = 4.6, Time 1 1 2 2 80 60 k2 = 8.2, h2 = 6.5, a2 = 2.3, b2 = 1.7, 40 20 Population Q1 0 0 2 4 6 8 10 12 14 16 18 20 and using the following migration rates Time 200 m 0 1 m 0 1 n 0 n 0 100 21 = . , 12 = . , 21 = , 12 = , Population P2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Time we have respectively the following stable equilibria 400 200

E = (8.0057,0,8.1962,0), E = (0,5.7,0,6.5). The Population Q2 6 11 0 0 0.05 0.1 0.15 0.2 0.25 0.3 separatrices are pictured in the top row of Figure 10, the Time rightb frame containing patch 1b and the left one patch 2. If we change the migration rates, allowing a faster return toward patch 1, Fig. 3: The equilibrium E4 is stable for the parameter values r1 = 0.15, r2 = 90, s1 = 0.55, s2 = 61, K1 = 250, K2 = 300, H1 = 120, H2 = 500,e a1 = 12, a2 = 0.06, b1 = 3, b2 = m21 = 0.1, m12 = 2.0, n21 = 0, n12 = 0, 0.015, m12 = 0, m21 = 0.9, n12 = 0, n21 = 0.8, with initial conditions (110,80,64,250). The equilibrium coordinates are the second equilibrium E11 remains unchanged, but we E4 =(0,0,205,2799,474,9398). nd instead that the point E6 = (9.3399,0,5.4726,0) has moved toward higher P and lower P population values. 1 b 2 e The separatrices are plottedb in the bottom row of Figure 10. It is also clear that the basins of attraction in patch 1 hardly change, while in patch 2 the basin of attraction of replacing some of the former ones. Granted that the population Q2 appears to be larger with a higher coexistence is once again forbidden for its instability, emigration rate from patch 2. Correspondingly, the one of three new equilibria arise, containing either one or both P2 becomes smaller in patch 2, according to what populations in the patch toward which migrations occur, intuition would indicate. leaving the other one possibly empty. The principle of competitive exclusion in this case may still occur at the metapopulation level, but apparently coexistence at 6.4 Unidirectional migrations equilibrium E4 might be possible in the patch toward which populations migrate if the stability conditions (12) When migrations are allowed from patch 1 into patch 2 coupled withe the feasibility conditions (11) are satised. only, a number of other possible equilibria arise, in part This appears to be also an interesting result.

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100 200 50

100 Population P1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Population P1 0 100 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Time 50

40 Population Q1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time 20 100 Population Q1 0 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Population P2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.08 Time 100 0.06

0.04 50 0.02 Population Q2

Population P2 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Time

200

100

Population Q2 Fig. 6: The two top populations P1 and Q1 occupy the 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time rst patch, the bottom ones P2 and Q2 the second patch. The coexistence equilibrium can be achieved at E16 = (18.4266,18.4266,18.6164,18.6164) for the parameter values Fig. 4: The equilibrium E is stable for the parameter values r = 8 1 r1 = 100, r2 = 100, s1 = 100, s2 = 100, K1 = 250, K2 =e 250, 148.8149, r = 95.9844, s = 121.9733, s = 171.8885, K = 2 1 2 1 H1 = 250, H2 = 250, a1 = 5, a2 = 5, b1 = 5, b2 = 5, m12 = 0, 228.8361, K = 223.9932, H = 201.4337, H = 216.7927, a = 2 e 1 2 1 m21 = 0.5, n12 = 0, n21 = 0.5. 71.2694, a2 = 47.1088, b1 = 68.1972, b2 = 7.1445, m12 = 0, m21 = 0.8175, n12 = 0, n21 = 0.5186.

200

100 Population P1 0 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time 10 10 Population P1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 Time Population Q1 0 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time 100 200 Population Q1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 100 Time Population P2 200 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time 100 40 Population P2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 20 Time Population Q2 0.06 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.04 Time

0.02 Population Q2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Fig. 7: The coexistence equilibrium E6 is stable for the parameter values r1 = 179.2820, r2 = 48.8346, s1 = 162.9841, s2 = Fig. 5: The equilibrium E12 is stable for the parameter values 27.9518, K1 = 202.4929, K2 = 265.3457, H1 = 204.9169, H2 = r1 = 70.3319, r2 = 117.0528, s1 = 183.4387, s2 = 151.4400, 203.4834, a1 = 58.2431, a2 = 69.7650, b1 = 94.4784, b2 = K1 = 219.0223, K2 = 207.e5854, H1 = 226.5399, H2 = 293.4011, 77.2208, m12 = 0.9758, m21 = 0.5674, n12 = 0.4716, n21 = a1 = 56.8824, a2 = 1.1902, b1 = 16.2182, b2 = 31.1215, m12 = 0.2537. 0, m21 = 0.2630, n12 = 0, n21 = 0.4505.

allowing a faster rate for the population P, we again nd Again exploiting the algorithm of [5], we investigated that the second equilibrium E11 is unaffected, but the rst also the change in shape of the basins of attraction of the one lowers its population values, becoming two equilibria E6 and E11, For this unidirectional E6 = (5,0,9.9907,0), see bottome row of Figure 11. In migrations model. Using once again the demographic this case the basins of attraction seem to have opposite parameters (20),e we take ate rst the migration rates as behaviors.e With a higher migration rate for P2, its basin of follows attraction in patch 2 gets increased, while in patch 1 becomes smaller. This result is in agreement with m21 = 0.1, m12 = 0, n21 = 0.1, n12 = 0, intuition, in patch 1 the P population become smaller and larger instead in patch 2. obtaining equilibria E6 = (7.875,0,8.3408,0) and E11 = (0,5.5860,0,6.6192). This result is shown in the top row of Figure 11, again patche 1 in the right frame and patche 2 6.5 Final considerations in the left one. Instead with the choice We briey discuss also the model bifurcations for the m21 = 2.4, m12 = 0, n21 = 0.1, n12 = 0, unidirectional migration model. If r1 < m21 and s1 < n21,

c 2015 NSP Natural Sciences Publishing Cor. 1750 D. Belocchio et. al.: On the Principle of Competitive Exclusion in...

10 10

40 9 9

8 8 20 7 7 Population P1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 6 6 Time 1 5 2 5 Q Q

200 4 4

100 3 3

Population Q1 2 2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1 Time 0 0 0 2 4 6 8 10 0 2 4 6 8 10 20 P P 1 2

10 10 10 Population P2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 9 9 Time 8 8

200 7 7

100 6 6 2 1 Population Q2 5 5 Q 0 Q 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time 4 4

3 3

2 2

1 1

0 0 Fig. 8: The coexistence equilibrium E11 is stable for the 0 2 4 6 8 10 0 2 4 6 8 10 P P 1 2 parameter values r1 = 46.2191, r2 = 191.5950, s1 = 70.5120, s2 = 171.4748, K1 = 240.3233, K2 = 256.7841, H1 = 244.9968, Fig. 10: Only population P is able to migrate: separatrix of the H = 263.8244, a = 49.1146, a = 43.3295, b = 77.5334, 2 1 2 1 basins of attraction of the equilibria E6 and E11 lying on the axes. b2 = 38.0149, m12 = 0.4620, m21 = 0.6463, n12 = 0.8896, n21 = The demographic parameters are given by (20). Right column: 0.8370. patch 1; left column: patch 2. Top: mb21 = 0.1,b m12 = 0.1, n21 = 0, n12 = 0. Bottom: m21 = 0.1, m12 = 2.0, n21 = 0, n12 = 0.

20

10 Population P1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time 10 10

20 9 9

10 8 8 Population Q1 0 7 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time 6 6

30 1 2 5 5 Q Q

20 4 4

10 3 3 Population P2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2 2 Time 1 1

40 0 0 0 2 4 6 8 10 0 2 4 6 8 10 P P 20 1 2 Population Q2 0 10 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time 9 9

8 8

7 7

6 6

1 5 2 5 Fig. 9: The two top populations P1 and Q1 occupy the rst patch, Q Q the bottom ones P and Q the second patch. The coexistence 4 4 2 2 3 3 equilibrium, E16 = (10.7367,10.7367,15.0240,15.0240), is 2 2 obtained just for a very specic parameter choice namely r = 1 1 1 0 0 0 2 4 6 8 10 0 2 4 6 8 10 P P 110, r2 = 80, s1 = 110, s2 = 80, K1 = 360, K2 = 270, H1 = 360, 1 2 H2 = 270, a1 = 10, a2 = 5, b1 = 10, b2 = 5, m12 = 0.5, m21 = 0.1, Fig. 11: Unidirectional migrations: separatrix of the basins n12 = 0.5, n21 = 0.1. of attraction of the equilibria E6 and E11 lying on the axes. Demographic parameters are given by (20). Right column: patch 1; left column: patch 2. Top: me21 = 0.1,e m12 = 0, n21 = 0.1, n12 = 0. Bottom: m21 = 2.4, m12 = 0, n21 = 0.1, n12 = 0. the only feasible equilibria are E2, E3, which are stable under the additional conditions s2 < b2K2 and r2 < a2H2. When r1 crosses the value m21 ande similarlye s1 n21, the two previous equilibria become unstable, and transcritica≥ l bifurcations give rise respectively to the equilibria E6 and E11. The equilibrium E4 may coexist with each one of the assumptions used to study congurations in patchy previous equilibria, but in this case E2 and E3 muste be environments may not always hold. Under suitable unstable,e whereas E6e and E11 may be stable if their conditions, competing populations may coexist if only stability conditions hold. e e one migrates freely, or if migrations for both populations In the two particulare casese above discussed, of just are allowed in the same direction and not backwards. This one population allowed to migrate and of unidirectional appears to be an interesting result, which might open up migrations, our analysis shows that the standard new research directions.

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c 2015 NSP Natural Sciences Publishing Cor. 1752 D. Belocchio et. al.: On the Principle of Competitive Exclusion in...

Davide Belocchio Alessandro Marchino received the Master degree received the Master degree in Mathematics at the in Mathematics at the University of Torino. University of Torino.

Roberto Cavoretto received the PhD in Ezio Venturino Mathematics at the University received the PhD in Applied of Torino. At present Mathematics at SUNY he is Assistant Professor, Stony Brook. Professor working on problems of Numerical Analysis, in approximation theory. in the editorial board of several international Journals. His present interests lie in mathematical biology.

Giacomo Gimmelli received the Master degree in Mathematics at the University of Torino.

c 2015 NSP Natural Sciences Publishing Cor.