Addendum to A Simple Differential Equation System for the Description of Competition among Religions

Transcription

1 International Mathematical Forum, Vol. 7, 2012, no. 54, Aenum to A Simple Differential Equation System for the Description of Competition among Religions Thomas Wieer Germany thomas.wieer@t-online.e Abstract We propose a linear ifferential equation system for the escription of competitions among populations (e.g. religions) for followers. The interaction among the populations is moele by the use of constant coefficients an, as the new feature, by aitional amping factors. Mathematics Subject Classification: 91C99 Keywors: population ynamics, opinion formation, religions, conversion, aherence, ifferential equation system 1 Introuction The growth an ecline of religions in terms of their populations is a complex social phenomenon. A central aspect is the competition of coexisting religions for followers. Unerstoo as quantitative objects, populations along with their increase an ecrease can be expresse in mathematical terms, in particular in the form of ifferential equations. Thus, it is tempting to try a escription of religious competition by a suitable set of ifferential equations. In fact, several moels have been evelope in the last years by researchers from various isciplines, see for example [1] (physics) or [2] (economics). Recently, we have propose such a moel in terms of ifferential equations [3] an we are fully aware of its simplicity compare to the complexity of the social reality. In the present paper we outline an alternative, in a sense even more simple moel. Consier a population of N iniviuals in total. How can the interactions among all these iniviuals an subsequent processes like group forming an group issolution be moele mathematically? A common approach is the use of an incience matrix N with elements n i,j where i, j = 1,..., N. Element n i,j inicates the relation between elements i an j, often one just has n i,j = 1 or 0 inicating whether the iniviuals know each other or not. Although

2 2682 Thomas Wieer N is a very powerful tool its use is hampere by the fast growing number of interactions N(N 1)/2. The opposite approach consists in the use of population numbers x i with R i x i = N an usually R N. Interactions between populations are then expresse by ifferential equations which contain proucts like x i x j. The most famous example in population ynamics for this metho is the prey-preator system. In [3] we outline such a ifferential equation system for the competition of two religions an one irreligious group. 2 The ifferential equation system In the present aenum to [3] we propose yet another approach. First of all, remember that the population x i is a function x i (t) of time t. The interaction in time of population x i (t) with population x j (t) takes place via some (cultural) mechanism. As a consequence, x i (t) can gain followers from x j (t) accoring to some gain coefficient ɛ i,j but this coefficient is ampe by a amping factor f(i). The amping factor (it coul be efine as an amplification factor in an alternative set-up) escribes in an overall manner the intensity of the mutual interaction among x i (t) an x j (t), its form is with f(i) = e ζ (x i (t) N(t)) N(t) (1) N(t) = x 1 (t) + x 2 (t)+,..., +x R (t) (2) for up to R ifferent religions. The coupling strength is expresse by the coupling factor 0 ζ. Note that is is always 0 f(i) 1 since x i N(t). Define the term T 0 (i, j) = 0 if i = j an T 0 (i, j) = 1 if i j an the opposite term T 1 (i, j) = 1 if i = j an T 1 (i, j) = 0 if i j. As in [3] we have terms α i which represent the growth rate of population i, but for simplicity here we omit eath rates. Then our new ifferential equation system DE is R R t x i (t) = (ɛ i,j f(i) T 0 (i, j) + α i T 1 (i, j)) x i (t) ɛ j,i f(j)t 0 (i, j) x j (t) (3) i=1 j=1 Let us restrict to the case of R = 3 interacting groups, like in [3]. Then we can write the DE (3) in explicit form as t x 1 (t) = ɛ 1,2 e ζ ( x2(t) x3(t)) x 1 (t) ɛ 2,1 e ζ ( x 1 (t) x 3 (t)) x 2 (t) (4) +ɛ 1,3 e ζ ( x 2 (t) x 3 (t)) x 1 (t) ɛ 3,1 e ζ ( x 1 (t) x 2 (t)) x 3 (t) + α 1 x 1 (t)

3 Aenum to: Competing religions 2683 t x 2 (t) = ɛ 2,1 e ζ ( x1(t) x3(t)) x 2 (t) ɛ 1,2 e ζ ( x 2 (t) x 3 (t)) x 1 (t) (5) +ɛ 2,3 e ζ ( x 1 (t) x 3 (t)) x 2 (t) ɛ 3,2 e ζ ( x 1 (t) x 2 (t)) x 3 (t) + α 2 x 2 (t) t x 3 (t) = ɛ 3,1 e ζ ( x1(t) x2(t)) x 3 (t) ɛ 1,3 e ζ ( x 2 (t) x 3 (t)) x 1 (t) (6) +ɛ 3,2 e ζ ( x 1 (t) x 2 (t)) x 3 (t) ɛ 2,3 e ζ ( x 1 (t) x 3 (t)) x 2 (t) + α 3 x 3 (t) 3 Numerical solutions The evolution of x i (t) with time t can be calculate numerically from 4, 5, 6. As in [3] we abet R 3 with the greatest growth rate: α 1 = 0.001, α 2 = 0.001, α 3 = Furthermore both R 2 an R 3 gain followers from R 1, R 3 has an aitional inflow from R 2, R 1 has no inflow: ɛ 2,1 = 0.01, ɛ 3,1 = 0.005, ɛ 3,2 = These coefficients are chosen arbitrarily for our present examples, they nee to be fitte to empirical ata for empirical stuies. In the present examples we just concentrate on the role of the coupling factor ζ. Figure 1 exhibits the case ζ = 0.0 for which no amping of the interaction occurs. One observes the rapi ecline of R 1 an contrarily the rise of R 3. Figure 2 shows the case ζ = 1.0 with strong amping of the interaction. In that case, R 1 survives. Figure 1: Solution to (4) - (6) for ζ = 0.0.

4 2684 Thomas Wieer Figure 2: Solution to (4) - (6) ζ = Discussion an Conclusion The present approach to the growth an ecline of religions in terms of their populations introuces the amping factors f(i) as a new element into the moeling by ifferential equations. These f(i) coul help to escribe (in a very broa manner) for example the epenency of the interactions on the local istributions of the populations. The actual form of the f(i) may be aapte to the concrete problem uner consieration. In its propose form, the equation system (3) is linear in the x i which is at least more easy to treat then a non-linear system with proucts like x i x j. Anyhow, further progress in the moeling will epen on the incorporation of empirical ata. 5 Appenix The following commans will set up the DE (4), (5), (6) within the computer algebra system Maple (version 13). > restart: > with(plots): with(detools): > T0 := proc (a, b) local Result; if a = b then Result := 0 else Result := 1 en if; Result en proc; T1 := proc (a, b) local Result; if a = b then Result := 1 else Result := 0 en if; Result en proc; > ien := 3; jen := ien; > NA := a(x[ummy](t), ummy = 1.. jen); > i := i ; j := j ; epsilon := epsilon ;

5 Aenum to: Competing religions 2685 alpha := alpha ; zeta := zeta ; envti := exp(zeta*(x[i](t)-na)/na); envtj := exp(zeta*(x[j](t)-na)/na); > for i to ien o DE[i] := iff(x[i](t), t) = a((epsilon[i, j]*envti*t0(i, j) +alpha[i]*t1(i, j))*x[i](t) -epsilon[j, i]*envtj*t0(i, j)*x[j](t), j = 1.. jen) en o; The following Maple commans will solve the DE (4), (5), (6 ) in numerical manner. > x1t0 := 750; x2t0 := 300; x3t0 := 50; > for i to ien o for j to jen o epsilon[i, j] := 0.1e-2 en o en o; > epsilon[2, 1] := 0.1e-1; epsilon[3, 1] := 0.5e-2; epsilon[3, 2] := 0.1e-1; > for i to ien o lt[i] := 0 en o; > alpha[1] := 0.1e-2; alpha[2] := 0.1e-2; alpha[3] := 0.2e-1; > zeta := 1.0; > ABPsys := [DE[1], DE[2], DE[3]]; > label1 := textplot([100, 270, x[1], font = [TIMES, 15]]); label2 := textplot([100, 460, x[2], font = [TIMES, 15]]); label3 := textplot([100, 680, x[3] ], font = [TIMES, 15]); aplot:=deplot(abpsys, [x[1](t),x[2](t),x[3](t)], t = , [[x[1](0)=x1t0,x[2](0)=x2t0,x[3](0)=x3t0]], scene=[t,x[1](t)], thickness=2, linestyle=1, linecolor=black, stepsize=1.0): bplot:=deplot(abpsys, [x[1](t),x[2](t),x[3](t)], t = , [[x[1](0)=x1t0,x[2](0)=x2t0,x[3](0)=x3t0]], scene=[t,x[2](t)], thickness=2, linestyle=2, linecolor=black, stepsize=1.0): pplot:=deplot(abpsys, [x[1](t),x[2](t),x[3](t)], t = , [[x[1](0)=x1t0,x[2](0)=x2t0,x[3](0)=x3t0]], scene=[t,x[3](t)], thickness=2, linestyle=3, linecolor=black, stepsize=1.0): isplay([aplot,bplot,pplot,label1,label2,label3]); References [1] Daniel M. Abrams, Haley A. Yaple, Richar J. Wiener: Dynamics of Social Group Competition: Moeling the Decline of Religious Affiliation, Physical Review Letters 107 (2011), [2] Rachel M. McCleary (e.), The Oxfor Hanbook of the Economics of Religion, Oxfor University Press, 2011.

6 2686 Thomas Wieer [3] Thomas Wieer: A simple ifferential equation system for the escription of competition among religions, International Mathematical Forum 6(8) (2011), , /wieerIMF pf. Receive: July, 2012