Advanced Studes n Bology, Vol. 3, 20, no. 7, 333-345 Predcton of Mgraton Path of a Colony of Bounded-Ratonal Speces Foragng on Patchly Dstrbuted Resources Rebysarah S. Tambaoan, Jomar F. Rabaante, Ramon Joseph P. Esteves and Mchael C. Vlladelrey Insttute of Mathematcal Scences and Physcs Unversty of the Phlppnes Los Baños Correspondence should be addressed to Jomar F. Rabaante, frabaante@up.edu.ph Abstract A theoretcal and conceptual model was formulated to determne, as well as to predct, possble mgratory route of a colony of bounded-ratonal speces. A network wth connected patches was set to represent the envronment, and the mgraton behavor of the colony was assumed to follow the Optmal Foragng Theory. Varous dynamc envronmental factors such as qualty and quantty of avalable food, dstance of the food sources, populaton growth rate of the colony, exstence of compettors, and cost of transferrng from one locaton to another can be consdered n choosng the next resdence of the speces. The preferences of speces were ntegrated usng fuzzy Analytc Herarchy Process. The predcted mgratory route was obtaned by successve Mxed-Integer Lnear Goal Programmng. The acqured result was treated only as a satsfcng soluton because of the speces cogntve lmtatons and possble conflctng preferences. Keywords: mgraton, optmal foragng theory, mult-crtera decson makng, lnear goal programmng
334 R. S. Tambaoan et al Introducton There are several reasons why speces mgrate [,4,9,]. In most cases, they mgrate to have the best possble lvng condtons. An ndvdual or a communty may move to a new locaton because of overpopulaton and competton n ther current habtat. Speces, such as nsects and humans, mgrate n response to the depletng resource quantty. In ths paper, we answered the queston What s the mgratory route that the members of the colony should collectvely choose to follow n order to obtan sustanng benefts wth mnmal cost? by formulatng a mathematcal model. In determnng the possble mgraton route, only a satsfcng soluton to our model was consdered because of the bounded ratonalty of the speces. A satsfcng soluton means that only the adequate crtera are satsfed, and ths soluton may not be optmal because of the exstence of conflctng preferences. Speces are bounded ratonal snce they have lmted knowledge about ther envronment. Makng decsons have always been a bg part of daly lfe, especally when makng choces among dfferent alternatves. The determnaton of alternatves and ther mportance s a crucal and essental step [2,3,7,2]. Ths study wll help n predctng the satsfcng mgraton path of speces by consderng ther preferences and needs for survval, gven that the behavor of the speces follows the model s assumptons. Moreover, t wll gve nsghts about the actual mgraton strategy of some speces by measurng the devaton of the actual mgraton route of the speces from the acqured results of the model. The degree of devaton depends on how ratonal the speces are. The necessary steps n the formulaton of our theoretcal and conceptual model are () determne and rank the preferences of speces, (2) determne a satsfcng mgraton route that would satsfy, or nearly satsfy, the colony s wants and needs consderng the carryng capacty of the envronment, (3) measure the underachevement of the colony s wants and needs, (4) determne how long the envronment can provde enough food for the colony s survval, and (5) perform necessary senstvty analyss after obtanng a soluton. The man assumptons of the model are the behavor of the colony follows the Optmal Foragng Theory and the speces forage on patchly dstrbuted food sources [8,3,4,5,6]. Optmal Foragng Theory mples that the speces consder the costs and benefts of foragng n a certan patch or nearby patches. Also, the tme perod between mgraton actvtes was assumed to be dscrete. The needed nput values are rankng of speces preferences, populaton growth rates, quantty of food n each food source, dstance of patches from one another, cost of competton, cost of mgraton, quantty of food needed by the colony, and natural producton and depleton rates of the food sources. Some, f not all, of these nput nformaton were assumed to be known to the colony. Obtanng these nput nformaton s feasble because some colones do have scouts tasked to explore ther surroundng envronment. However, only the known
Predcton of mgraton path 335 food sources, new resdence locatons and compettors can be consdered n the model. 2 Fuzzy Analytc Herarchy Process Analytc Herarchy Process (AHP) s a mult-crtera decson makng technque that s commonly used n rankng dfferent alternatves. Saaty s scale [5,0,7] for the par-wse comparson between alternatves s llustrated n Table. The comparson matrx showng the par-wse comparson among alternatves s of the followng form: where s the relatve mportance of alternatve over alternatve quantfed usng Saaty s scale. Table. Saaty s scale for par-wse comparson. Saaty s scale Relatve mportance between factors Equally mportant 3 Moderately mportant 5 Strongly mportant 7 Very strongly mportant 9 Extremely mportant 2,4,6,8 Intermedate values To address the ambguty n each preference made, the comparson matrx s transformed nto a fuzzy par-wse comparson matrx usng the followng relatonshp [6]: a r a. Table 2. Scale for fuzzy par-wse comparson. Fuzzy scale membershp values (r ) Relatve mportance between factors 0.5 Equally Important 0.5 0.55 0.6 Slghtly Important 0.6 0.65 0.7 Important 0.7 0.75 0.8 Strongly Important 0.8 0.85 0.9 Very Strongly Important 0.9 0.95.0 Extremely Important
336 R. S. Tambaoan et al The weghts w, w2,..., wn are determned by the formula: b w where, b n. n b n r Some of the preferences may be nconsstent so t s suggested to measure these nconsstences to be able to mprove the udgments made. The consstency of the comparson matrx can be checked usng the consstency rato [0]. 3 Model Formulaton A network s consdered where the possble new resdence locaton wth connected food patches comprses the set of nodes. We wll call the locaton of the resdence as hve. There are four possble types of nodes: () old hve; (2) food source only; (3) possble new hve and a food source; and (4) possble new hve but not a food source, whch s llustrated n Fgure. Fgure. Network of the food sources and the possble new locaton Several dynamc envronmental factors can be consdered n choosng the next hve locaton, such as preferences of the speces. The preferences of the speces wth respect to the factors consdered are ncorporated n the model and are ranked usng fuzzy Analytc Herarchy Process (AHP). The populaton growth rate and food consumpton of the mgratng colony and the compettors can also be taken nto account. The actual dstance between two nodes s converted nto ts equvalent perceved (nverse) dstance value. In ths paper, we used the negatve exponental functon as the perceved dstance functon. The colony has ts lmtaton on the dstance t can travel. There s a maxmum foragng dstance on whch they can only travel. In addton, the values used n the computatons were non-dmensonalzed and normalzed usng nfnty-norm.
Predcton of mgraton path 337 The soluton to the mathematcal program generated can be solved usng Operatons Research software. Defnton Varables and Parameters for tme perod t number of nodes number of factors or alternatves except the dstance between nodes and the amount of food avalable at a node set of all possble new hve locaton set of nodes wthn foragng dstance ndcator of possble new hve locaton; ths s our decson varable where,f node s chosen X 0,otherwse normalzed perceved (nverse) dstance functon value from node node normalzed total quantty of food avalable n node normalzed value or amount of factor at node (excludng the dstance between nodes and the amount of food avalable at a node) weghts derved from fuzzy AHP assgned to factor aggregate contrbuton of factors to the qualty of node, where v m w F k k k number of survvors of the mgratng colony after a competton total number of survvors of all the competng colones after the competton utlty functon value assgned to node based from the cost of competton the least number of survvors n order for a colony to be consdered a wnner n a competton the most number of survvors n order for a colony to be consdered a loser n a competton cost n wnnng a competton (same dmenson as ) cost n losng a competton (same dmenson as ) cost n a ted competton (same dmenson as ) mnmum quantty of food needed by the colony (same dmenson as ) penalty weght for the underachevement of penalty weght for the cost of mgraton costs of mgratng to node (asde from the factors ncluded n the computaton of, ths may nclude the rsk of exstence of predators near the possble hve locaton and the energy needed to transfer from the old hve to the new hve) to
338 R. S. Tambaoan et al amount of remanng food at node from the prevous tme perod expected total amount of food that the speces can forage n node consderng competton devaton varable correspondng to the overachevement of devaton varable correspondng to the underachevement of The Mxed-Integer Lnear Goal Program The goal of the model s to maxmze the beneft subect to the costs assocated wth the mgraton of the speces. The am s to maxmze the Obectve Functon: n X ( Y ) The obectve s subect to the followng constrants. Constrant : n X where f node s not an element of Constrant 2: For node, If there s no competton at node, Y n c q v Else, f there s competton at node, Y n c U where Sq v U P( S s2)[ q v C] P( S s) C2 P( s S s2) C3 ST Constrant 3: For, v m w F k k k Constrant 4: ( X c R d d ) A H J where Sq P( S s2) q P( s S s2), f there s a compettor at node R S T q, otherwse The frst constrant restrcts the speces n choosng exactly one new hve locaton. The second constrant ncorporates the factors consdered n choosng d
Predcton of mgraton path 339 the new locaton. The dstance of a food source from the hve and the quantty of avalable food are assumed to be the most mportant factors n the speces decsons. In the thrd constrant, the preference of the speces s beng ncorporated accordng to ts mportance. The possble compromse between the avalablty of food and the dstance travelled by speces durng foragng s reflected on the fourth constrant. The non-negatve varables and are the devatonal varables that represent the devaton above and below (mnmum quantty of food needed by the colony), respectvely. When there s known competton n the food source, correspondng cost s ncorporated. If the colony wns the competton wth the probablty gven by, then colony can take the entre food source on the locaton but wll pay for the cost of competton. On the other hand, f the colony loses wth probablty gven by, the colony wll not get the food source and wll pay for the cost of competton. But there could also be a possblty that the competng colones wll coexst, ths means that they can share the food source on the locaton but both colony stll need to pay for the cost of competton. 4 Prototype Example Consder the network n Fgure 2. The dstances between nodes are gven. Assume that the equvalent perceved dstance functon ( ) of the actual dstance ( ) s equal to. The perceved dstance functon value can also be nterpreted as the energy needed durng travel. Old Hve Fgure 2. Network of possble locatons and food sources
340 R. S. Tambaoan et al The avalable quantty of food n every node s gven n Table 3. Let,, and the maxmum flght dstance equal to. The dmenson of should be the same as the dmenson of 's. Assume that one food s good to sustan an ndvdual member of the colony. The correspondng values for the cost of mgraton are gven n Table 4. Dfferent mgratng costs were consdered n every mgraton tme perod because the energy needed by the colony to transfer from the old hve to the new hve vares. The dmenson of s the same as the dmenson of. In ths example, we assumed the value of depends on the value of. Table 3. Avalable amount of food per node. Node 2 3 4 5 6 7 Amount of food 6 3 0 4 0 7 Table 4. The cost of mgraton at a gven tme. Tme 0 2 3 4 5 45% Y 30% Y 35% Y 40% Y 50% Y 50% Y There are three factors that were par-wse compared. The factors are sucrose concentraton, extent of smell and brghtness of color of the food. The gven par-wse comparson matrx s shown n Table 5. These qualtatve factors were quantfed accordngly. The values of each factor at every node are gven n Table 6. Table 5. The par-wse comparson matrx of the factors. Factor Factor 2 Factor 3 Factor 6 4 Factor 2 /6 /3 Factor 3 /4 3 Table 6. Value of Factor at node. Node Node 2 Node 3 Node 4 Node 5 Node 6 Node 7 Factor 4 3 4 0 3 0 3 Factor 2 0 6 0 3 5 6 7 Factor 3 6 4 8 0 5 0 5 Assume that there s a compettor at node 5 and node 7. The correspondng utlty functon nput values are gven n Table 7. The dmensons of, and are always the same as the dmenson of s.
Predcton of mgraton path 34 Table 7. Utlty Functon Input Values. P(S 30) P(S 20) P(20<S<30) C C 2 C 3 S/S T Node 5 0.6 0.5 0.25 0.4 qv 2.5 qv.2 0.6 qv Node 7 0.58 0.4 0.28 0.3 qv.5 qv 0.08 qv 0.6 Suppose that the natural producton and depleton of the food source at tme perod can be computed by consderng the perodc formula:, where s the natural producton/depleton of the food source n node at tme perod, s the maxmum producton of the food source, and s the remanng food n node at tme perod. Also, suppose that the populaton growth of the speces follows the formula: where s the colony s populaton at tme perod and s the colony s populaton at tme perod. After consderng the gven prototype example and conductng necessary calculatons, the mathematcal program for the model has been formulated. In the comparson made between the dfferent factors consdered usng fuzzy AHP, the resultng weghts are, and. For the tme perod, the resultng mathematcal program s as follows: Maxmze 0.2498 X2 0.323942 X4 0.04359 X5 0.68403 X6 0.47623 X7 d subect to: X X X X X 0.55697 X 0.6762 X 0.3564 X 0.325 X 0.76009 X d d.36 X {0,}, 2, 4,5,6,7 d, d 0, For the tme perod, the resultng mathematcal program s as follows: Maxmze 0.3295 X 2 0.455083 X 4 0.06877 X5 0.243584 X6 0.23287 X7 d subect to: X X X X X 0.56589 X 0.73606 X 0.43270 X 0.35093 X 0.8693 X d d.64 X {0,}, 2, 4,5,6,7 d, d 0,
342 R. S. Tambaoan et al For the tme perod, the resultng mathematcal program s as follows: Maxmze 0.246509 X2 0.374003 X 4 0.06588 X5 0.220907 X6 0.22357 X7 d subect to: X X X X X 0.44726 X 0.6435 X 0.42394 X 0.33897 X 0.88053 X d d.96 X {0,}, 2, 4,5,6,7 d, d 0, For the tme perod, the resultng mathematcal program s as follows: Maxmze 0.3684 X2 0.47237 X 4 0.079 X5 0.29732 X6 0.5784 X7 d subect to: X X X X X 0.75495 X 0.9022 X 0.73383 X 0.5385 X 0.76357 X d d 2.36 X {0,}, 2, 4,5,6,7 d, d 0, For the tme perod, the resultng mathematcal program s as follows: Maxmze 0.69049 X2 0.32784 X4 0.0899 X5 0.260222 X6 0.73674 X7 d subect to: X X X X X 0.37405 X 0.70992 X 0.83763 X 0.52435 X 0.8283 X d d 2.83 X {0,}, 2, 4,5,6,7 d, d 0, For the tme perod, the resultng mathematcal program s as follows: Maxmze 0.058483 X 2 0.47054 X 4 0.0899 X5 0.207822 X6 0.60277 X7 d subect to: X X X X X 0.227 X 0.3072 X 0.76006 X 0.4075 X 0.6304 X d d 3.39 X {0,}, 2, 4,5,6,7 d, d 0,
Predcton of mgraton path 343 In the obectve functons, some values of s are negatve, whch s an mplcaton of the competton exstng n a node or of the hgh mgraton cost. The soluton to the gven prototype example s llustrated n Fgure 3 and tabulated n Table 8. Table 8. Soluton to the gven prototype example. t Preferred new hve (node) Underachevement of food requrement Obectve Functon Value 0 4 0.68840-0.3666 4 0.90390-0.452 2 7.07950-0.8586 3 4.45780-0.9896 4 4 2.200 -.8006 5 7 2.75990-2.605 Old Hve Fgure 3. Vsualzaton of the soluton to the gven prototype example The result depcts that the colony wll not transfer to a new hve locaton untl the end of tme perod because ther present locaton (old hve, node 4) stll gves the most satsfcng benefts. But after tme perod, the colony wll mgrate to node 7, however, eventually the colony wll mgrate back to node
344 R. S. Tambaoan et al 4 at and. At, the colony wll choose agan node 7. After tme perod, the speces could no longer fnd a patch where they can sustan the food requrement they needed. Though node 2 s a possble new hve and a food source, the colony cannot mgrate to that locaton because t does not provde enough beneft for the colony. Snce the colony s consderng the flght dstance t can only travel as well as the dstances of food sources, then t s possble that t cannot mgrate to a specfc new locaton. The negatve value of the obectve functon value can be translated as the shortage on the desred benefts of speces. The value of the devatonal varable sgnfes the shortage from the mnmum amount of food needed by a colony. These shortages could also be nterpreted as the amount that the envronment lacks n provdng the colony s need. Senstvty analyss can be done to observe varaton n the optmal soluton as nput values vary. 5 Summary and Recommendaton The study amed to determne a satsfcng mgraton path of a colony of speces foragng on patches. The study consdered varous factors such as the colony s collectve preferences, avalablty of food, qualty and quantty of food, dstance of the food sources, populaton growth rate, exstence of competton and cost of transferrng from one locaton to another. The behavor of the colony was assumed to follow the Optmal Foragng Theory, where the colony consders the costs and benefts assocated durng mgraton. The preferences of speces durng mgraton were ranked usng fuzzy Analytc Herarchy Process (AHP). Successve Mxed-Integer Lnear Goal Programmng was used n makng the necessary analyss and fndng the satsfcng soluton. To llustrate the method for determnng a specfc mgratory route for a certan colony of speces, a prototype example was demonstrated. In makng the model, several thngs should be examned and revewed. Frst and most mportantly are the assumptons that are taken nto account. Next thng s the lstng of the factors and rankng t accordngly usng the fuzzy AHP algorthm. In ratng the preferences of speces durng mgraton, t s a must to check the consstency of the udgments made. Moreover, before dong the calculatons, all values should have been non-dmensonalzed and normalzed. It s recommended to add more factors and an actual observaton or expermentaton be done n order to test the valdty of the model. The conceptual model formulated can be modfed to ft a specfc bologcal problem. References [] B. Hoare, Anmal Mgraton: remarkable ourneys n the wld, Unversty of Calforna Press, 2009.
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