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УДК 517.9
Impact of Reflexivity on a Dynamics of a Population with Optimal Migration and Global Information Access
Michael G. Sadovsky* Mariya Yu. Senashova^
Institute of Computational Modeling SB RAS Akademgorodok, 50/44, Krasnoyarsk, 660036
Russia
Received 10.05.2015, received in revised form 10.06.2015, accepted 20.07.2015 Comparison of reflexive vs. non-reflexive behaviour is provided, for the dynamics of globally informed beings in a community with optimal migration.
Keywords: population dynamics, optimization migration, global information access, Ferchulst equation. DOI: 10.17516/1997-1397-2015-8-3-340-342
1. Reflexive-free model of the dynamics of globally informed beings
We consider a dynamics of spatially distributed communities where being perform optimal relocation of themselves in space; optimality means maximization of net reproduction [1, 2]. We shall suppose a community to consist of two stations (habitation entity) occupied with two species; migration is stipulated to be a transfer from station to station, exclusively. Besides, the beings are supposed to have global access to information/knowledge on environmental conditions, subpopulation density, transfer cost etc. The aim of the paper is to figure out the impact of reflexivity on the dynamics.
We start from a single species population subdivided into two subpopulations. Let the dynamics of each subpopulation in migration-free case follows Verchult's equation:
Nt+i = Nt (a - b • Nt); Mt+i = Mt (c - d • Mt) , (1)
where Nt and Mt are the abundances of the subpopulations in the time moment t, in station [T]
and | II |, respectively; a and c are Malthusian parameters while b and d describe the density dependent regulation. The functions in the parentheses in (1) are the net reproduction, in the corresponding stations.
A migration (between the stations) runs if and only if the living conditions at the immigration station become better than these latter at the habitation station, with respect to transfer cost p. The figure p, (0 < p < 1) is a probability of a successful transfer from station to station; i. e. the transfer that brings no harm for further reproduction.
The migration from station ]T] to station II (and from II to [T], respectively) starts, if:
a - bNt <p ■ (c - dMt) or c - dMt <p ■ (a - bNt) . (2)
* msad@icm.krasn.ru t msen@ icm. krasn. ru © Siberian Federal University. All rights reserved
The number A of beings migrating from station to station must equalize the inequality (2). Obviously, A ^ Nt (or A ^ Mt). If the migration runs at the time moment t, then A is defined by
• fAr a - bNt + pc - pdMt \ <- f c - dMt + pa - pbNt \
A = miMNt'-b^-A = min(M-^-}• (3)
So, the model works as following: for each t a migration conditions (2) are checked, and they hold true, the migration fluxes (3) are determined. Then the upgraded abundances Nt and Mt are determined as
Nt = Nt + pA, Mt = Mt - A; or Nt = Nt - A, Mt = Mt + pA
in dependence on the migration direction. It may be At = 0, for some t. Finally, the next generation abundances for t +1 time moment are calculated according to (1), with upgraded figures Nt and Mt. The model (1-3) is irreversible in time: migration is a projection here. It should be stressed, that the model has no reciprocal migration fluxes.
2. Reflexive model of the dynamics of globally informed beings
Reflexivity means an ability to foresee the behaviour of a counterparty at the conflict [3]. We stipulate here that each subpopulation is divided into two "subspecies": reflexive (Rt) and reflexive-free NRt beings. Suppose that N0 = NRq, M0 = Rq1 for t = 0. Further, the abundances in each station are the sums of these latter of both "subspecies": NR^+R^ and NR^+R11. The model is to be changed in the following way to define the migration conditions:
a - b(NRl + R1) < p ■ (c - d(NRlJ + R^)) orc - ¿(NRf1 + R1/) < p ■ (a - b(NR[ + Rt1)) • (4)
A feasibility of a condition in (4) yields the migration flux ANR for reflexive-free subspecies accompanied with the upgrade of abundances NRf and NR^1: NR f = NR f - A,
NRl1 = NR 1 1 + pA, (I ^ II) or TVR = NR 1 + pA, NRf = NRf1- A, (// ^ I). Then the constraints (4) is checked, for reflexive subspecies, and the migration flux Ar is determined together with the relevant upgraded abundances. Finally, the abundances for both subspecies, at the time moment t + 1 are calculated: NRf+i = NRl (a - b ■ (NVR + RJ^, r/+i = R/ ^ - b .(NVR + Rt)); NRt+i =
NVR? (a - b ■ (N + R^)) , Rt+1 = f (c - d ■ (N + ff )) .
3. Simulations
We simulated the dynamics with the following parameters: 2 ^ a ^ 10 and 2 ^ c ^ 10 both with the step equal to 0.16; 0.0001 < b < 0.0005 (same for d) with the step equal to 0.000008, and 0.1 < p < 1 with the step 0.1. Each run was as long, as 103 generations; initial abundances of all subspecies were equal to 103, in both stations.
So, for every figure of p the set of 504 parameters a, c, b, d figures has been used in simulation to calculate the average abundances: N is the average over the generations series over two stations abundance of reflexive-free subspecies, and NR is similar figure for reflexive species. Totally, 10 x 504 trajectories of the model have been simulated, and the average figures have been determined. An advantage of reflexivity has been observed for some parameters combinations, while an opposite situation for other; there were some combinations where advantage of neither strategy has been observed. Table at the left shows this situation: SN is the portion of cases where reflexive-free strategies bring advantage, SNR is the portion of the opposite cases; finally, S is the portion of cases with the parity of strategies.
References
[1] A.N.Gorban, M.G.Sadovsky, Optimal strategy of spatial distribution: Alle's effect, Rus. J. of Gen. Biol., 50(1989), no. 1, 16-21.
[2] M.G.Sadovskii, M.Yu.Senashova, On the problem of modeling reflexive behavior in conflict by an example of biological communities, J. of Appl. and Industr. Math., 8(2014), no. 3, 375-384.
[3] V.A.Lefebvre, The law of self-reflexion, Reflexive Processes and Control 1(2002), no. 2, 91-99.
Влияние рефлексивного поведения на динамику численности одновидовой популяции глобально информированных особей
Михаил Г. Садовский Мария Ю. Сенашова
В статье сравниваются модели динамики популяции пространственно распределённой одновидовой популяции глобально информированных особей с рефлексивным поведением и без него.
Ключевые слова: динамика популяции, оптимизационная миграция, глобальная информированность, уравнение Ферхюльста.
p Sn ,% Snr, % S,%
0.1 31.10642655 48.9710452 19.92252825
0.2 13.52659247 65.56794783 20.9054597
0.3 6.37460658 77.29145911 16.33393431
0.4 4.098623796 83.98300407 11.91837213
0.5 4.335381006 86.07933738 9.585281614
0.6 5.963787304 86.8060692 7.230143498
0.7 8.569575901 86.01201558 5.41840852
0.8 10.47916745 86.42967115 3.091161406
0.9 12.47091839 86.483895 1.045186614
1.0 8.030887804 91.78426439 0.184847806
