A Fuzzy Differentil Approch to strong Allee Effect bsed on the Fuzzy Extension Principle XAVIER BERTRAN 1, NARCIS CLARA 2, JOAN CARLES FERRER 1 Universitt de Giron 1 Deprtment d Empres, 2 Deprtment d Informàtic i Mtemàtic Aplicd Cmpus de Montilivi s/n, 1771 Giron SPAIN xvier.bertrn@udg.edu, nrcis.clr@udg.edu, joncrles.ferrer@udg.edu Abstrct: The Allee effect is relted to those spects of dynmicl of popultions connected with decresing in individul fitness when the popultion size diminishes to very low levels. In this wor we propose fuzzy pproch to Allee effect tht permits to del with uncertinty. This fuzzy proposl is bsed on the extension principle nd shows different behviour of the equilibrium points with regrd to the crisp model. Key Words: Allee Effect, Fuzzy Number, Level Sets, Extension Principle, Fuzzy Differentil Eqution 1 Introduction In single popultion dynmics the rte of chnge of the number of individuls x(t) is often expressed s function of the per cpit rte of chnge ϕ(x) such s dx = xϕ(x) (1) dt The Mlthus eqution considers constnt per cpit growth nd is the simplest wy to model the growth of specie. This eqution hs only one equilibrium point t x =nd the popultion tends to infinity if > ( is unstble) or vnishes if < ( is stble). This point of view implies n exponentil growth of the popultion. The logistic eqution dx dt = g(x) =rx(1 x ) (2) solves the question of the existence of positive equilibrium point becuse x = is n symptoticlly stble point nd the size of the popultion tends to. Figure 1 describes this very well nown eqution. One feture of Eqution (2) often criticized in ecologicl dynmics is tht for smll vlues the rte growth is lwys positive. Severl equtions hve been proposed to del with this problem. We will focus our ttention to the strong Allee effect wht implies tht bellow positive number the popultion goes to extinction. A differentil eqution tht governs the popultion growth ting into ccount these spects is dx dt = f(x) =rx(1 x )(x ) (3) Eqution 3 hs three equilibrium points, two re stble ( nd ) nd one is unstble (). Figure 1 shows the behviour of function f nd its differences with the logistic eqution. dx/dt m g Figure 1: Logistic (g) nd Allee (f) functions The per cpit growth of the logistics eqution is liner decresing function insted of the Allee effect is incresing for smll vlues lthough negtive. This remr is reflected in Figure 2 where v nd u re the per cpit growth of the logistic nd Allee effect respectively. The Allee effect hs strong evidence in ecologicl systems ([7]) nd is ssocited to wide rnge of systems focusing in conspecific interctions, rrity nd niml socility ([2, 3, 19, 2]). Conspecific interction cn help to understnd ggregtive behviour ([18]), Allee effect cn explin some proprieties bout the distribution of individuls between ptches ([11]), it is obviously relted to extinction ([16]) nd influ- f M x ISSN: 1792-4235 111 ISBN: 978-96-474-199-1
dx/xdt u nd the level sets (or α-levels with α [, 1]) of the fuzzy number f( x(t)) = ỹ(t) te the nown form ỹ(α, t) =[y(α, t), y(α, t)] with v x y(α, t) =min{f(s) s [x(α, t), x(α, t)]} y(α, t) =mx{f(s) s [x(α, t), x(α, t)]} (5) Figure 2: Percpit growth for logistic (v) nd Allee (u) equtions ence ny system of mximum sustinble yields s some fisheries ([13]). Bidimensionl models lso benefit with the introduction of the Allee effect ([1]). 2 Fuzzy nlysis bsed on the extension principle Environmentl nd economicl systems re usully subject to uncertinty which ffects the mthemticl structure of them. References deling with theoreticl or pplied ([17]) fuzzy differentil equtions show gret number of subjects of interest nd points of view. J.J. Bucley et l, P Dimond nd J.J. Nieto del with first order differentil equtions ([5, 9, 1, 15]), Bucley et l nd D.N. Georgiou et l with high order differentil equtions ([6, 12]) nd for more pplied focus relted to this wor we cite ([4, 8, 14]). In our context we must consider tht the popultion is given by fuzzy rel number x(t) for every t identifying the symbol of its membership function with itself in order to simplify the nottion, so stisfying Normlity: s R such tht x(t)(s )=1 x(t) is convex x(t) is upper-continuous The support of x(t), [ x(t)] = {s : x(t)(s) > } is compct set Let f : R R be rel function. Given fuzzy number x R then the fuzzy extension f of f following the Zdeh s extension principle ([21]) is defined s f( x)(s )= { sup { x(s) } if f 1 (s ) = s f 1 (s ) (4) if f 1 (s )= In order to simplify the nottion we note x(α, t) =x 1 nd x(α, t) =x 2. It is simple to show tht for the Mlthus fuzzy eqution nd > the trjectories tend to infinite. On the other wy, if <the trjectories tend to the line x 1 =. For the fuzzy logistic eqution J. J. Nieto et l divided the study of the trjectories in three regions showing complex behviour. As generl rule the trjectories leve one region nd enter into nother region, but cn lso tend to infinite. This behviour depends on the initil conditions ([14]). In the cse of the Allee effect the utonomous eqution hs three equilibrium points nd s consequence increses the number of regions. Moreover, in most cses it is not possible to get n explicit expression of the trjectories nd only n implicit expression is possible. All tht mes more complex the study of the fuzzy Allee effect. The rel function f(x) ttints minimum t m (we note f(m) =p) between nd nd mximum t M (we note f(m) =q) between nd following the expressions obtined from f (x) =. Applying the Zdeh s extension principle given by (4) six cses rises depending on x 1 nd x 2 belong to the zones Z 1 = [,m], Z 2 = (m, M] or Z 3 =(M,+ ). This zones led to the study of six regions R 1 = {(x 1,x 2 ) R 2 : x 1 Z 1 x 2 Z 1 } R 2 = {(x 1,x 2 ) R 2 : x 1 Z 1 x 2 Z 2 } R 3 = {(x 1,x 2 ) R 2 : x 1 Z 1 x 2 Z 3 } R 4 = {(x 1,x 2 ) R 2 : x 1 Z 2 x 2 Z 2 } R 5 = {(x 1,x 2 ) R 2 : x 1 Z 2 x 2 Z 3 } R 6 = {(x 1,x 2 ) R 2 : x 1 Z 3 x 2 Z 3 } The fuzzy first order differentil eqution is trnsformed to bidimensionl system of differentil equtions which vribles re x 1 nd x 2. In the next section we nlyze the trjectories of this system. ISSN: 1792-4235 112 ISBN: 978-96-474-199-1
3 Solution of the Fuzzy Allee eqution From the definition of the system it is cler tht the solutions depend on which region the initil conditions belong. When it is possible we get the solutions, if not, we obtin the trjectories nd explin their behviour. We consider s initil conditions x 1 (t )=x 1 nd x 2 (t )=x 2. 3.1 Solution of x = f( x) in region R 1 In this cse f(x 1 ) f(x 2 ) so [x 1,x 2 ]=[f(x 2),f(x 1 )] = x 1 = r x 2( x 2 )(x 2 ) x 2 = r x 1( x 1 )(x 1 ) (6) We get non liner system of differentil equtions. Dividing both equtions nd integrting we obtin the implicit solution of the trjectories: x 4 1 4 ( + )x3 1 3 + x2 1 2 = = x4 2 4 ( + )x3 2 3 + x2 2 2 + C (7) We obtin the vlue of C imposing the initil conditions C = x4 1 4 ( + )x3 1 3 + x2 1 2 x4 2 4 + +( + ) x3 2 3 x2 2 2 There re different possibilities for the behviour of the trjectories tht we summrize s follows Behviour strting t Region R 1 If x 1 = x 2 then the trjectories tend to the point (, ). If x 1 =nd x 2 > then the trjectories leve the region R 1. Otherwise the trjectories tend to points belonging to the ordintes xe. 3.2 Solution of x = f( x) in region R 2 [x 1,x 2]=[m, mx{f(x 1 ),f(x 2 )}] We need to distinguish two cses. Cse A: f(x 1 ) f(x 2 ) x 1 = p x 2 = r x 1( x 1 )(x 1 ) Solving the trjectories we get x 4 1 4 ( + )x3 1 3 + x2 1 2 = p r x 2 + C Applying the initil conditions we obtin C = x4 1 4 ( + )x3 1 3 + x2 1 2 p r x 2 Behviour strting t Region R 2 (Cse A) (8) If x 1 = then the trjectories leve the region. If x 2 = nd x 1 > then the trjectories tend to the point (,). Otherwise some trjectories tend to the ordintes xe nd others enter the region R 1. Cse B: f(x 1 ) <f(x 2 ) x 1 = p x 2 = r x 2( x 2 )(x 2 ) (9) which is n uncoupled system of differentil equtions. Solving it x 1 = pt + C 1 (x 2 ) (1) x 2 ( x 2 ) = C 2e r ( )t Imposing the initil conditions we obtin C 1 = x 1 pt ( x2 ) ( x2 ) e r( )t C 2 = x 2 x 2 Finlly, the trjectory is ( (x2 ) x 2 x 1 =ln ( x 2) ) x + x 1 2 (x 2 ) ( x 2 ) Behviour strting t Region R 2 (Cse B) If x 1 = then the trjectories leve the region R 2. If x 2 = M then the trjectories leve the region R 2 nd enter into the region R 3. Otherwise the trjectories eep on the region nd some of them tend hed for the ordintes xe or to the line x 2 = M. ISSN: 1792-4235 113 ISBN: 978-96-474-199-1
3.3 Solution of x = f( x) in region R 3 [x 1,x 2] = [min{f(x 1 ),f(x 2 )},q] We need lso to distinguish two cses. Cse A: f(x 1 ) <f(x 2 ) [x 1,x 2 ]=[p, q] x 1 = pt + C 1 (11) x 2 = qt + C 1 nd imposing the initil conditions we obtin C 1 = x 1 pt nd C 2 = x 2 qt The trjectory is line with negtive slope x 2 = q p x 1 + px 2 (1 + q)pt x 1 p Cse B: f(x 1 ) f(x 2 ) [x 1,x 2]=[f(x 2 ),q] x 1 = r x 2( x 2 )(x 2 ) x 2 = q (12) Clculting the trjectories we get x 4 2 4 ( + )x3 2 3 + x2 2 2 = q r x 1 + C Applying the initil conditions we obtin C = x4 2 4 ( + )x3 2 3 + x2 2 2 q r x 1 Behviour strting t Region R 3 If x 1 = then the trjectories leve the region R 3. Otherwise the trjectories tend to the line x 1 =. 3.4 Solution of x = f( x) in region R 4 [x 1,x 2 ]=[f(x 1),f(x 2 )] x 1 = r x 1( x 1 )(x 1 ) x 2 = r x 2( x 2 )(x 2 ) (13) which is uncoupled system of differentil equtions. Solving it (x i ) x i ( x i ) = C ie r( )t i =1, 2 Imposing the initil conditions for i =1, 2 ( xi ) ( xi ) e r( )t C i = x i x i Behviour strting t Region R 4 If x 1 = x 2 <then the trjectories tend to the point (m, m). If x 1 = x 2 >then the trjectories tend to the point (M,M). If x 1 = nd x 2 >then the trjectories tend to the point (, M). If x 1 <nd x 2 then the trjectories eep on the region. Some of them hed on to the line x 1 = m nd the others to the line x 1 = x 2. If x 1 <nd x 2 then the trjectories eep on the region. Some of them hed on to the line x 1 = m nd the others to the line x 2 = M. If x 1 >nd x 2 >then the trjectories eep on the region. Some of them hed on to the line x 1 = x 2 nd the others to the line x 2 = M. 3.5 Solution of x = f( x) in region R 5 [x 1,x 2 ] = [min{f(x 1),f(x 2 )},q] We need to distinguish two cses. Cse A: f(x 1 ) <f(x 2 ) x 1 = r x 1( x 1 )(x 1 ) x 2 = q (14) Clculting the trjectories we get (x 1 ) x 1 ( x 1 ) = Ceqr 1x 2 Applying the initil conditions we obtin ( x1 ) ( x1 ) e qr C = 1 x 2 x 1 x 1 Behviour strting t Region R 5 (Cse A) If x 1 <then the trjectories tend to the line x 1 = m. If x 1 >then the trjectories tend to the line x 1 = M. ISSN: 1792-4235 114 ISBN: 978-96-474-199-1
Cse B: f(x 1 ) f(x 2 ) x 1 = r x 2( x 2 )(x 2 ) x 2 = q (15) which is n uncoupled system of differentil equtions. Solving it x 4 2 4 ( + )x3 2 3 + x2 2 2 = q r x 1 + C Imposing the initil conditions we obtin C = x4 2 4 ( + )x3 2 3 + x2 2 2 q r x 1 Behviour strting t Region R 5 (Cse B) If x 2 <then the trjectories tend to the line x 1 = m. If x 2 >then the trjectories tend to the line x 1 = M. will follow similr procedure for the study of the fuzzy Alle effect. The crisp differentil eqution hs three equilibrium points: e 1 =, e 2 = nd e 3 =. Looing t the Figure 1 it is obvious tht e 1 nd e 3 re stble (symptoticlly) nd e 2 is unstble. In the fuzzy context the differentil eqution is converted to system of two crisp differentil equtions depending on where belong x 1 nd x 2. Solving the homogenous system we find three equilibrium points E 1 = (, ), E 2 = (, ) nd E 3 = (, ) which belong to the region R 1, R 4 nd R 6 respectively. We note F (x 1,x 2 )=(x 1,x 2 ). 4.1 Study of the point E 1 As this point belongs to the zone R 1 F (x 1,x 2 )=(f(x 2 ),f(x 1 )) nd [x 1,x 2]=[f(x 2 ),f(x 1 )] = we get 3.6 Solution of x = f( x) in region R 6 This cse is nlyticlly identicl to the cse 3.1 but the behviour of the trjectories is different. Behviour strting t Region R 6 If x 1 = x 2 <then the trjectories tend to the point (, ). If x 1 = x 2 >then the trjectories tend to the point (, ). If x 1 x 2, x 1 <nd x 2 <then the trjectories tend to the line x 1 = x 2 or to the line x 2 =. If x 1 nd x 2 >then the trjectories tend to the line x 1 = M. If x 1 >nd x 2 >then the trjectories tend to the line x 1 = or to the line x 1 = x 2. 4 Qulittive study of the Fuzzy Allee eqution The qulittive study of the fuzzy Mlthus eqution for > shows the sme behviour tht the crisp becuse there is n unstble equilibrium point. On the other wy, if < there is n unstble point unlie the crisp pproch in which cse the equilibrium point x =is stble. The qulittive study of the fuzzy logistic eqution mde by J. J. Nieto shows tht there re two equilibrium points tht, surprising, re unstble ([14]). We =[ r x 2( x 2 )(x 2 ), r x 1( x 1 )(x 1 )] Linerizing the system round the equilibrium point we obtin ( ) r J(F (, )) = r Solving the seculr eqution we find its eigenvlues λ 1 = r nd λ 2 = r, hence (, ) is sddle point. 4.2 Study of the point E 2 As this point belongs to the zone R 4 F (x 1,x 2 )=(f(x 1 ),f(x 2 )) nd [x 1,x 2 ]=[f(x 1),f(x 2 )] = we get =[ r x 1( x 1 )(x 1 ), r x 2( x 2 )(x 2 )] Linerizing the system round the equilibrium point we obtin ( ) r( ) 1 J(F (, )) = r( ) 1 Solving the seculr eqution we get its eigenvlues λ 1 = λ 2 = r( ) 1 > hence (, ) is n unstble node. ISSN: 1792-4235 115 ISBN: 978-96-474-199-1
4.3 Study of the point E 3 As this point belongs to the zone R 6 F (x 1,x 2 )=(f(x 2 ),f(x 1 )) nd [x 1,x 2]=[f(x 2 ),f(x 1 )] = we get =[ r x 2( x 2 )(x 2 ), r x 1( x 1 )(x 1 )] Linerizing the system round E 3 we obtin ( ) r( ) J(F (, )) = r( ) Similrly t the previous cses we find the eigenvlues λ 1 = r( ) > nd λ 2 = r( ) <, hence (, ) is sddle point. [9] P. Dimond, Brief note on the vrition of constnts formul for fuzzy differentil equtions, Fuzzy Sets nd Systems, 129, 22, pp. 65 71. [1] P. Dimond, P. Kloeden, Metric Spces of Fuzzy Sets. Theory nd Applictions, World Scientific Publishing Co. 1994, [11] S.D. Fretwell, Popultions in Sesonl Environment, Princenton University Press 1972, pp. 27 33. [12] D.N. Georgiou, J.J. Nieto, R. Rodríguez López, Initil vlue problems for higher-order fuzzy differentil equtions, Nonliner Anlysis, 63, 25, pp. 587 6. 5 Conclusion The study of the strong fuzzy Allee effect under uncertinty from the point of view of fuzzy extension principle shows tht fuzziness chnges the behviour of the set of solutions becuse ll equilibrium points re unstble unlie the crisp cse. References: [1] P Aguirre, E. González-Olivres, E. Sáez, Three limit cycles in Leslie-Gower predtor-prey model with dditive Allee effect, SIAM J. Appl. Mth., 69(5), 29, pp. 1244 1262. [2] W.C. Allee, The Socil Life of Animls, Willim Heinemnn 1938, pp. 27 33. [3] P. Amrsere, Allee effects in metpopultion dynmics, Am. Nt., 152, 1998, pp. 298 32. [4] L.C. Brros, R.C. Bssnezi, P.A. Tonelli, Fuzzy modelling in popultion dynmics, Ecologicl Modelling, 128, 2, pp. 27 33. [5] J.J. Bucley, T. Feuring, Fuzzy differentil equtions, Fuzzy Sets nd Systems, 11, 2, pp. 43 54. [6] J.J. Bucley, T. Feuring, Fuzzy initil vlue problem for N th-order liner differentil equitons, Fuzzy Sets nd Systems, 121, 21, pp 247 255. [7] F. Courchmp, T. Clutton-Broc, B. Grenfell, Inverse density dependence nd the Allee effect, Trends Ecol. Evol., 14, 1999, pp. 45 41. [8] E.Y. Deeb, A. de Korvin, E.L. Koh, On fuzzy logistic difference eqution, Differentil Equtions Dynmicl Systems, 4, 1996, pp. 149 156. [13] R. Lnde, S. Engen, B.E. Sether, Optiml hrvesting, economic discounting nd extinction ris in fluctuting popultions, Nture, 372, 1994, pp. 88 9. [14] J.J. Nieto, R. Rodríguez López, Anlysis of logistic differentil model with uncertinty, Int. J. Dynmicl Systems nd Differentil Equtions, 1(3), 28, pp. 164 177. [15] J.J. Nieto, R. Rodríguez López, D. Frnco, Liner first-order fuzzy differentil equtions, Internt. J. Uncertin, Knowliedge-Bsed Systems, 14(6), 26, pp. 687 79. [16] J.C. Poggile, From behviourl to popultion level: growth nd competition, Mth. Comput. Model., 27, 1998, pp. 41 49. [17] I.G. Rtiu, C. Crste, L.P. Doru Plese, M. Boscoinu, Fuzzy modeling economics, Proceedings of the 9th WSEAS Interntionl conference on simultion, modelling nd optimiztion, 29, pp. 51 56. [18] J.M. Reed, A.P. Dobson, Behviourl constrints nd conservtion biology: conspecific trction nd recruitment, Trends Ecol. Evol., 8, 1993, pp. 253 256. [19] P.A. Stephens, W.J. Sutherlnd, Consequences of the Allee effect for behviour, ecology nd conservtion, TREE, 14(1), 1999, pp. 41 45. [2] W.J. Sutherlnd, The importnce of behviourl studies in conservtion biology, Anim. Behv., 56, 1998, pp. 81 89. [21] L.A. Zdeh, Fuzzy Sets, Informtion nd Control, 8, 1965, pp. 338 353. ISSN: 1792-4235 116 ISBN: 978-96-474-199-1