N I nigjrj lLY

As the mapping in increasing, for each C < 0 there exists z = z(C) e (0, R) such that C+(l-£)^)2e(R-r) <0 if re [0,z(C))t while

C+(l-£)(^)2^(i?-r)>0 if re [z(C),R). Moreover, C z(C) is decreasing and lim z(C) = R, limz(C) = 0. (83)

Thus, proceeding as above, at r = R inequality (82) reduces to

_(1_£)<_(1_£)P, which is satisfied because 1 — e < 1 and p > 1. By continuity, there exists 5 = 8(e) > 0 such that (82) holds in [R - 5,R). Moreover, by (83), there exists C < 0 such that z(C) = R — 5. For this choice of C, ip is a subsolution of (80). By construction,

Moreover, choosing a sufficiently large A gives

Thus, by the uniqueness of the solution L(x), we have that, for each e > 0,

^(|x-£o|) <L(x) < i>e(\x - Z0|) , \x-x0\< R, and, therefore, by (84), we find that l-e< lim < 1 + e■ (85)

As (85) holds for each sufficiently small e > 0, we obtain that lim = (86)

Clearly, (79) follows readily combining (86) with Theorem 3.3, which concludes the proof if fi = Br(xq). Now, suppose

Then, setting

L(x) := i/>(r), r = \x — xq\, where ip is the unique solution of

Now, consider the function where Rm (Ri + i?2)/2. Arguing as above, it is easy to check that, for each sufficiently small e > 0, there exists a constant Ae > 0 such that, for any A > As, the function

$£(r):=A + (l + e)0(r), r€(Ri,Ri), is a positive supersolution of (87). Similarly, there exists C < 0 for which the function

is a non-negative subsolution of (87). Thus, thanks to Theorem 3.2 of J. Lopez-Gomez16, (87) possesses a positive solution, say ijje, in between

and for sufficiently large A. Consequently, by the uniqueness of rp(r), as a consequence of Theorem 1.1, we find that ip < ip < V>£ and, hence,

As (88) holds true for any sufficiently small e > 0, (86) holds. Theorem 3.3

ends the proof in this case as well. This concludes the proof. □

References

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2. C, Bandle and M. Marcus, Large solutions of semilinear elliptic equations: Existence, uniqueness and asymptotic behavior, J. D'Analyse Math. 58 (1991), 9-24.

3. C. Bandle and M. Marcus, On second order effects in the boundary behavior of large solutions of semilinear elliptic problems, Diff. Int. Eqns. 11 (1998), 23-34.

4. F. C. Cirstea and V. Radulescu, Uniqueness of the blow-up boundary solution of logistic equations with adsorbtion, C.R. Acad. Sci. Paris, Ser. I, 335 (2002), 447-452.

5. F. C. Cirstea and V. Radulescu, Asymptotics for the blow-up boundary solution of the logistic equation with absorption, C. R. Acad. Sci. Paris, Ser. I, 336 (2003), 231-236.

6. Y. Du and Q. Huang, Blow-up solutions for a class of semilinear elliptic and parabolic problems, S I AM J. Math. Anal. 31 (1999), 1-18.

7. J. Garcia-Meliân, R. Letelier-Albornoz and J. C. Sabina de Lis, Uniqueness and asymptotic behaviour for solutions of semilinear problems with boundary blow-up, Proc. Amer. Math. Soc. 129 (2001), 3593-3602.

8. D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin 1977.

9. R. Gômez-Renasco and J. Lopez-Gômez, On the existence and numerical computation of classical and Non-classical solutions for a family of elliptic boundary value problems, Nonl. Anal. T.M.A. 48 (2002), 567-605.

10. V. A. Kondratiev and V. A. Nikishin, Asymptotics, near the boundary, of a solution of a singular boundary value problem for a semilinear elliptic equation, Diff. Eqns. 26 (1990), 345-348.

11. J. B. Keller, On solutions of Au = f(u), Comm. Pure Appl. Math. X (1957), 503-510.

12. M. Marcus and L. Veron, Uniqueness and asymptotic behavior of solutions with boundary blow-.up for a class of nonlinear elliptic equations, Ann. Inst. Henri Poincarél4 (1997), 237-274.

13. C. Loewner and L. Nirenberg, 1974, Partial differential equations invariant under conformai or projective transformations, in Contributions to Analysis, p. 245-272, Academic Press, New York, 1974.

14. J. Lopez-Gomez, Large solutions, metasolutions, and asymptotic behaviour of the regular positive solutions of sublinear parabolic problems, El. J. Diff. Eqns. Conf. 05 (2000), 135-171.

15. J. Lopez-Gomez, Dynamics of parabolic equations. From classical solutions to metasolutions, Diff. Int. Equations. 16 (2003), 813-828.

16. J. Lopez-Gomez, The boundary blow-up rate of large solutions, J. Diff. Eqns. 195 (2003), 25-45.

17. J. Lopez-Gomez, Global existence versus blow-up in superlinear indefinite parabolic problems, Sc. Math. Jap. 61 (2005), 493-517.

18. J. Lopez-Gomez, Metasolutions: Malthus versus Verhulst in Population Dynamics. A dream of Volterra, in Handbook of Differential Equations: Stationary Partial Differential Equations (M. Chipot and P. Quittner eds.), Elsevier. In press.

19. J. Lopez-Gomez, Optimal uniqueness theorems and exact blow-up rates of large solutions, J. Diff. Eqns. Submitted.

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COOPERATION AND COMPETITION, STRATEGIC ALLIANCES, AND THE CAMBRIAN EXPLOSION*

Departamento de Matemática Aplicada, Universidad Complutense de Madrid,

28040-Madrid, Spain E-mail: [email protected]

MARCELA MOLINA-MEYER Departamento de Matemáticas, Universidad Carlos III de Madrid,

28911-Leganés, Madrid, Spain, E-mail: [email protected]

Dedicated to the memory of J. Esquinas

Ecology, Economy and Management require a huge interdisciplinary effort to ascertain the hidden mechanisms driving the evolution of communities and firm networks. This article shows that strategic alliances in competitive environments provoke an explosive increment of productivity and stability through a feedback mechanism promoted by cooperation, while competition causes segregation within cooperative profiles. Some further speciation and radiation mechanisms enhancing innovation, facilitated by environmental heterogeneities and specific market regulations, might explain the biodiversity of life and the high complexity of industrial and financial markets. Extinctions occur by the lack of adaptation of strongest competitors to sudden environmental stress.

1. Introduction

This article uses some very recent advances in the theory of nonlinear Partial Differential Equations (PDEs) to show that combining cooperative and competitive interactions in spatially heterogeneous environments promotes

"Work partially supported by grant REN2003-00707 of the Spanish Ministry of Science and Education.

Ill persistence and a substantial increment of productivity, diversity and stability in Ecology and Economy7-3'21'44'18'41-10. The mathematical analysis of these issues is necessary to maintain existing communities and design optimal production strategies. To predict and evaluate these phenomena, we use a model combining hybrid interactions with spatial dispersion, based on the competition paradigm of Lotka28 and Volterra43, whose analysis is imperative to ascertain the balance of cooperation and competition. Very simple prototype models of this type were actually used by Black & Scholes6 and Merton31 to evaluate stock options; in such models, dispersion and distribution rates are inter-exchanged by volatility rates.

The most striking prediction from the mathematical analysis carried out here is that strategic alliances, even at a very localized level, can provoke a huge added value (overyielding) after some period of time if the strength of the alliance is sufficiently high. As the benefits of strategic alliances between competing firms tend to remain hidden, by fiscal reasons, and the periods of time necessary for revealing the real impact of localized cooperative interactions in ecosystems tend to be extraordinarily large —as suggested by the time span necessary to recover diversity in tropical ecosystems after mass extinctions1'20—, using mathematical models is imperative for evaluating the effects of very localized, but possibly strong, alliances in competitive environments. This article shows that these alliances can indeed provoke an explosive increment of productivity, but eventually at the price of segregation of the "weaker competitor" within cooperation lines (patches). The underlying segregation mechanisms might help to explain the extraordinary diversity of life and industrial and financial markets, through some further adaptation mechanisms to specific market regulations or spatial heterogeneities. Basically, competition provokes segregation, whereas cooperation promotes huge added values through a feedback mechanism that will be discussed later. The mathematical theory necessary to conduct this analysis is very recent25'26.

2. The Cambrian explosion

544 millions years ago there were three animal phyla with their corresponding variety of external forms, though 538 millions years ago there were thirty-eight; the same number that exists today, except for one or two extinctions34. Consequently, the vast existing diversity of body architectures appeared during a five-million-year interval beginning 543 millions year ago. This phenomenon is referred to as the "Cambrian explosion", since the first fossils of that period of time where found in the Cambrian Hills in Wales by Sedgwick who name it as the "Cambrian". The prior time span is called the Precambrian. Cambrian explosion supports that the history of life consists of long time intervals of "micro-evolution" combined with short "macro-evolution" periods, which have been the most prolific generators of biodiversity; in strong contrast with the classical theory of Darwin and Wallace, according to which evolution occurred gradually. Parker34 defends the theory that Cambrian explosion was originated by the sudden evolution into eyes —in less than one million years— of many of former skin energy receptors.

The architecture of eyes alone can provide a lot of information on how animals lived. For instance, the position of the eyes on the head alone can reveal the position of the animal in the chain food. Eyes positioned at the sides of the head, facing sideways, can scan wide angles and detect any movement from nearly all direction, tend to belong to preys, while eyes positioned together at the from of the head, facing forward, generally belong to predators. Eyes of predators are extremely better for pinpointing targets and estimating the distance between them, even though see less part of the surroundings.

As a consequence of vision, the complexity of the interactions between species grew drastically, promoting all kind of mutual competition, cooperation and predating strategies, that originated the extraordinary biodiversity of the Earth's biosphere in the blink of an eye. Therefore, the Cambrian explosion supports the theory that combining cooperative and competitive interactions in communities provokes explosive increments of productivity and diversity.

3. Cooperation and competition in Ecology

Although ecologists realize that cooperative interactions arise very often in nature, they are not paid the deserved attention in ecological studies yet. However cooperation was recognized as a crucial driving force in population dynamics by the most pioneering ecologists9,2, during the last decades many ecologists focused their efforts towards the understanding of competition in communities, as, within the paradigm of Darwin and Wallace, yet competition is thought to be the most important process governing communities evolution and biodiversity38,39,40,41,42. Although cooperative interactions are well documented between organisms from different kingdoms, as they can make significant contributions to each other's needs without sharing the same resources, documenting cooperative interactions between similar organisms in the abundance seems to be a hard task in empirical studies, because they do not arise in isolation but in combination with competition, which validates the "abiotic stress hypothesis" of Bertness and Callaway4. According to it, the importance of cooperation in plant communities increases with abiotic stress or consumer pressure8'5. Alternatively, the importance of competition increases when abiotic stress is low.

Besides the abiotic stress hypothesis seemingly contradicts the huge number of cooperative mechanisms observed in tropical ecosystems45, it seems rather controversial to think of cooperation as the main driving force governing evolution during mass extinctions1, when abiotic stress suddenly grew in a geological time scale —by whatever cause, unimportant here. Actually, the huge number of "Lazaro" species (those disappearing from fossil registers whose persistence is confirmed after some period of time) documented by zoo-paleontologists after mass extinctions rather supports the theory that most of the species tend to segregate into small refuge areas (deep see waters, caves, islands...) where they safely stay until environmental stress decreases. Therefore, competition seemingly is the main driving force governing evolution after mass extinctions, instead of cooperation, even though local cooperation does play a crucial role for avoiding massive extinctions, as it becomes apparent from the mathematical analysis carried out here. As a matter of fact, extinction rules tend to reverse during mass extinction periods, up to provoke extinction of the most powerful competitors during background extinctions —less adapted to afford sudden environmental stress, e.g., Nautilus and mammals, apparently weaker than ammonites and dinosaurs, could successfully pass through several mass extinctions, while amonites and dinosaurs did not1,20. Consequently, the main interaction patterns governing evolution in the abundance tend to be rather different from those in harsh environmental conditions.

Elton's hypothesis12 that greater diversity causes greater productivity (the rate of biomass production) and stability (low susceptibility to invasion), further strengthened by Pimentel35 and Margalef29, remains among the most controversial principles in plant communities of the last decades. Although most of empirical studies have confirmed it18,41, yet pure competition models have not been able to corroborate it30'42 — rather naturally. Although the available competition models in heterogeneous environments predict that coexistence leads to overyielding (phenomenon explaining why several competing species have greater productivity than any individual species in monoculture) and that overyielding has a strong stabilizing effect on total community biomass, the mathematical problem of validating Elton's hypothesis remains utterly open42. Quite strikingly, yet local cooperative interactions have not been incorporated into the mathematical discussion of the diversity-stability debate. As a consequence of the mathematical analysis carried out here, it is apparent that overyielding caused by local cooperative interactions is substantially higher than overyielding promoted by sheer competition, and that stability indeed increases, which might definitively validate the Elton hypothesis mathematically. Local cooperative interactions have remained outside the diversity-stability debate during the last decades because of the tremendous number of mathematical difficulties in searching for the combined effects of spatial heterogeneities, dispersion, cooperation and competition.

4. Cooperation and competition in Management

Most of industrial, commercial and service sector firms make strategic alliances of rather competitive nature. These alliances have typically two effects. Besides they allow sustaining the "weaker competitor", that otherwise would become extinct, they do increment productivity and benefits in a rather substantial way. Such hybrid interactions are utterly apparent by considering a number of recent alliances in the automobile sector (among others, BMW-Chrysler, PSA-Toyota, Honda-Isuzu, and OpelRenault-Suzuki). Most of these agreements focus on cooperation in Research & Development and manufacturing of one or several components or product lines, while distribution typically remains competitive. These alliances are widely known under the label of "allied in costs, rival in markets". The importance of strategic alliances in competitive markets has been so much recognized in the late nineties that the concept of "co-opetition" has emerged in management literature to name the hybrid behavior cooperation-competition7'22'16. Undoubtedly, strategic alliances combined with competition have shown to be an extraordinary mechanism for value creation, even though not paid the deserved attention until very recently, still being an under-researched theme.

Whether or not co-opetition strategies may be maintained for either short of long time intervals, experience dictates that only if extended during a sufficiently large period of time proves to be really helpful for the creation of knowledge and economic value. Actually, the added value of co-opetition during short periods of time may be only a small fraction more than the one provided by sheer competition10.

Extrapolating these ideas to an ecological context readily there emerge some of the main difficulties in isolating hidden local cooperative interactions between competitive species. Indeed, after all documented mass extinctions most of oceanic and terrestrial systems recovered in around one million years, except for the notable exception of tropical ecosystems, which needed many million years to attain their former biodiversity levels1. Thus, it is necessary to give the appropriate time to a co-opetition system up to attain its maximal overyielding, or added value. Such time might depend upon the strength of strategic alliances in a rather hidden way. Therefore, searching for general principles governing the dynamics of co-opetitive systems from empirical studies might be almost impossible, since a long time span might be necessary to attain them. Consequently, using mathematical models in co-opetition studies seems certainly imperative.

Ecology and Economy are closely related fields, as they are nothing more than a miraculous combination of matter and energy, even though humans still think of themselves in a Platonic way they are at the center of the Universe! Adopting this perspective, strategic alliances among firms might provide an extraordinary empirical model to analyze the balance between cooperation and competition in life communities. Firms are well acquainted that benefits (ecological overyielding) can blow-up as a result of an appropriate alliance, even though real figures are not usually published.

5. The importance of mathematical models

The importance of mathematical models relies upon the fact that they provide an idealized behavior against which reality can be judged and measured, promoting further interdisciplinary studies. The main paradigm of this being Newton synthesis in Physics. Newton introduced the concept of derivative and, hence, the differential equations to obtain the planetary laws of Keppler. Consequently, the Newtonian potential was a combined consequence of the intensive empirical studies of Brache and his predecessors, the exquisite mathematical taste of Keppler, and the tremendous audacious and genius of Newton. Bringing together the three methodologies promoted one of the most important scientific revolution of Physics.

Actually, in many circumstances mathematical models have shown to be imperative for making predictions that on the empirical side are "unpredictable" . Among them, the discovery of Hertztian waves from Maxwell equations, the prediction of the curvature of light by Einstein, or the pre diction of the existence of black holes by Hawking and Penrose.

6. Modeling cooperation in competitive environments

Subsequently, we adopt the language of population dynamics, though most of our findings should have a counterpart in Economics and Management Science. To analyze the effects of local alliances in the dynamics of competing species we think of two populations U and V randomly dispersing in the inhabiting area Q C RN, N > 1, according to the Fourier-Fick-Fisher's law. Precisely, we assume the evolution to be governed by the system of PDEs f dtU - DiV2{7 = \U (1 - K^U - K^a12V) \ dtV - D2V2V = nV (1 - K^V - K^anU) 1 '

where Di > 0, £>2 > 0 measure the dispersion rates of U, V, dt stands for time derivation, and V2 is the spatial dispersion operator — the Laplacian19,17,33'32. Typically, for each location x G Q and time t > 0, U(x, t) and V(x, t) measure the density of the populations, A and /x are the intrinsic growth rates of U and V, and K\ > 0, K2 > 0 are their carrying capacities. The functions ai2(x) and a2i(x) measure the nature and strength of the interactions. In competition models, ai2(x) > 0 and a2i(x) > 0 for each x G ft. The non-spatial counterpart of system (1) (D\ = D2 = 0) goes back to Lotka28 and Volterra43. The spatial models where introduced by Fisher15, to describe the spread of a novel alele, by Skellam37, to study invasion problems, and by Black & Scholes6 and Merton31, to evaluate stock options. Random dispersion entails the organisms to move from densely populated areas to less populated regions36, thought such tendency can change as a result of competition and cooperation. Setting b = (\K2/iiKi) «12, c = (liKi/\K2) a2i,

U=(K1/\)u, V = (K2/fi)v, system (1) can be written as

\ dtv — D2S/2v = \iv — v2 — cuv where we will focus our attention. This system must be completed with initial conditions u(x,0) = uo(x), v(x,0) = v0(x), (3)

for each x G ft (uq and vo are the initial population distributions), and some appropriate boundary conditions along the habitat edges, called dQ.

In this work we assume the region outside fi is immediately lethal and, hence, we impose u(x, t) = v(x, t) = 0 (4)

for every x £ dfl and t > 0, i.e., fi is surrounded by an absorbing boundary. Terrestrial-aquatic edges are absorbing boundaries for seeds of plant species incapable of surviving in both habitats14, as is the legislative boundary of Yellowstone National Park for bison dispersing into Montana, where they are shot to control the spread of brucellosis11. In model (2)-(4), the

Figure 1. An admissible spatial configuration of interactions. The species compete in Q^ n fij and cooperate in fl^ n il 1, whereas u predates on v in fl^ n fij, v predates on u in fîjj" and u (resp. v) is free from any action of v (resp. u) in (resp.

coefficient functions b = b(x) and c — c(x) provide the nature and strength of the interactions at every location x £ Cl, which is of hybrid type, since b(x) and c(x) are arbitrary functions. Precisely, we suppose that fi consists of iijj", the region where b(x) > 0 (u receives aggressions from v), the region where b(x) < 0 (u is facilitated by v), and where b(x) = 0 (u is free from v). In particular, is a refuge area (Figure 1). Similarly, the sign of c(x) divides fl into where v receives aggressions from u, fl~, where v is facilitated by u, and where v is free from u (Figure 1). These hybrid models, within the spirit of Volterra43 and Lotka28, go back to Lopez-Gomez & Molina-Meyer23'25.

7. The balance of cooperation and competition

As occurs in classical non-spatial models, in many circumstances the dynamics of (2)-(4) is regulated by its steady states, i.e., by the non-negative solution pairs (u, v) of

-£>iV2u = Au — u2- b(x)uv ^ —D2V2v = ¡jlv — v2 — c(x)uv (5)

Besides the "trivial state" (0,0), problem (5) admits three types of nonnegative solutions. Namely, the solutions having one component positive and the other vanishing, (u, 0) or (0, v), refereed to as the "semi-trivial positive solutions", and the solutions having both components positive, the so called "coexistence states". By well known results in nonlinear PDEs, (5) has a semi-trivial positive solution of the form (it, 0) if and only if

In such case, (u\, 0) is the unique among such states, where u\ > 0 stands for the unique positive solution of

(see Lopez-Gomez23). Subsequently, given a dispersion rate D > 0, a patch Pcfi, and a "potential" V in P, we denote by a[-DV2 + V; P] the "principal eigenvalue" of —£>V2 + V in P subject to zero boundary conditions on the edges dP37. Condition (6) provides the critical A, or D\, for il to maintain the species u in the absence of v. Similarly, (5) has a semi-trivial positive solution of the form (0, v) if and only if fi > a[—D2V2; D] and, in such case, (0,«^) is the unique among these solutions; v^ being the unique positive solution of f -D2V2v = fiv -v2 in f2 \ v = 0 on dfl

The states u\ and equal the asymptotic profiles of u and v in isolation as 11 oo.

Subsequently, we regard to A and // as the main parameters of the model. Analyzing the linearized stability of the states (0, and 0) reveals that

A = f(^):=a[-D1W2 + bv^n} H = 5(A) := (T[-D2V2 + cuA; il] are their curves of change of stability in the (A, ¿t)-plane. Setting

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