Influence of age polyethism on longevity of workers in social insects
Appendix

Adam Tofilski

Bee Research Department
Agricultural University
29 Listopada 52
31-425 Krakow, Poland
rotofils@cyf-kr.edu.pl

This is a Mathematica 4.0 notebook.

1 Simplified model

It is assumed that there are two sets of tasks, A and B, which are associated with aging-independent mortality rates mA and mB respectively. A fixed proportion of time f is devoted to A-type tasks. A worker cannot spend more than the maximum resource k available for the whole life. The rates of resource expenditure during tasks A and B are cA and cB respectively. In the simplified model aging does not affect the mortality of workers until a certain age is reached, when resources become exhausted. At this time all workers die. In the model the expected longevity of workers in colonies with and without age polyethism is compared. If there is no polyethism the workers perform tasks A and B in turn. If age polyethism is present the workers perform A-type tasks first and after time ts they switch to B-type tasks.

1.1  General case

Error messages are switched off.

Off[General::spell1]
Off[General::unfl]
Off[Solve::ifun]
Off[NIntegrate::nlim]

The expected longevity of workers in colonies without polyethism is given by

p0 = (∫_0^((k/(cA f  +  cB (1 - f)))))Exp[-(mA f + mB (1 - f)) t] \[DifferentialD]t
p0 = Simplify[%]

The expected longevity of workers during A-type tasks is given by

aA = (∫_0^ts) Exp[-mA t] \[DifferentialD]t
aA = Simplify[%]

The expected longevity of workers during B-type tasks is given by

aB = 
      Exp[-mA ts] (∫_0^(((k - ts cA)/cB)))Exp[-mB t] \[DifferentialD]t
aB = Simplify[%]

The expected longevity of workers in colonies with age polyethis given by

pa = aA + aB;

The switching time ts cannot be found in a standard way.

Solve[(aA/(aA + aB)) == f, ts]

1.2  Solution

It was noticed that the expected longevity of workers in colonies with age polyethism is the same as in colonies without polyethism when

cB = ((mB cA)/mA);

The expected longevity of workers in colonies without polyethism is given by

p0s = (∫_0^((k/(cA f  +  cB (1 - f)))))Exp[-(mA f + mB (1 - f)) 
              t] \[DifferentialD]t
p0s = Simplify[%]

The expected longevity of workers during A-type tasks is given by

aAs = (∫_0^ts) Exp[-mA t] \[DifferentialD]t
aAs = Simplify[%]

The expected longevity of workers during B-type tasks is given by

aBs = (E^(-mA ts)) (∫_0^(((k - ts cA)/cB)))Exp[-mB t] \[DifferentialD]t
aBs = Simplify[%]

When cB = ((mB cA)/mA) the switching time ts can be found analytically.

Solve[(aAs/(aAs + aBs)) == f, ts]
Simplify[%]

It can be demonstrated that the expected longevity of workers in colonies with age polyethism is the same as in colonies without polyethism when cB = ((mB cA)/mA).

pas = aAs + aBs /. %[[1]]
pas = Simplify[%]

p0s == pas

1.3  Numerical solution

The way in which the problem was solved does not imply that there is only one solution. Therefore the equations were solved numerically here.

Numerical values are assigned to parameters.

cA = 2;
cB = 4;
k = 200;
f = 0.5;
mA = 0.01;

Approximate function tsi represents the switching time ts.

tsv[(mB_)?NumberQ]  :=  
      ts  /. FindRoot[(aA/(aA + aB)) == f,  {ts,  5}]
tsi =  
        FunctionInterpolation[tsv[mB], {mB,  0.001,  0.06}];

Switching time ts is replaced with approximate function tsi.

pa = aA + aB /. ts ->  tsi[mB];

The difference between expected longevity of workers in colonies with and without age polyethism is plotted.

Plot[pa - p0, {mB,  0.001,  0.06}, 
        AxesLabel -> {"mB", "pa-p0"}];

1.4  Absence of aging

If there is no aging (cA = cB = 0) the maximum life span is infinity.

The previous definition of the parameters have to be removed to allow analytical computation.

cB =. 
cA =. 
cB =. 
k =. 
f =. 
mA =.

The expected longevity of workers in colonies without polyethism  is given by

p00 = (∫_0^∞) 
            Exp[-(mA f + mB (1 - f)) 
              t] \[DifferentialD]t
p00 = 
      Simplify[% /. {Re[f mA + mB - f mB] > 0 -> True}]

The expected longevity of workers during A-type tasks is given by

aA0 = (∫_0^ts) 
            Exp[-mA t] \[DifferentialD]t
aA0 = Simplify[%]

The expected longevity of workers during B-type tasks is given by

aB0 = 
      Exp[-mA ts] (∫_0^∞) 
              Exp[-mB t] \[DifferentialD]t
aB0 = % /. {Re[mB] > 0 -> True}

When there is no aging the switching time ts can be found analytically.

Solve[(aA0/(aA0 + aB0)) == f, ts]
Simplify[%]

When there is no aging the expected longevity of workers in colonies with age polyethism is the same as in colonies without polyethism.

pa0 = aA0 + aB0 /. %[[1]]
pa0 = Simplify[%]

p00 == pa0

2 More realistic model

It is assumed that mortality rate r increases exponentially with age t: r=m+ α(t^β) where m is aging-independent mortality. Aging-related mortality is described by two parameters: α controlling the magnitude of this mortality, and β controlling the shape of the curve depicting changes of aging-related mortality with age. α associated with tasks A and B is αA and αB respectively. Tasks A and B are associated with the same β.

The expected longevity of workers during A-type tasks is given by

aAx = (∫_0^ts) 
              Exp[-mA t - ((αA (t^(β + 1)))/(β + 1))] \[DifferentialD]t;

The expected longevity of workers during B-type tasks is given by

aBx = (∫_0^∞) 
              Exp[-mB (t + 
                      tm) - ((αB (((t + tα))^(β + 1)))/(β + 1))] \[DifferentialD]t;

The correction terms tm and tα adjust the survivorship at the beginning of B-type tasks period to that the end of A-type tasks period.

Solve[Exp[-mB tm] == Exp[-mA ts], 
      tm]
tm = tm /. %[[1]] /. 
        Log[Exp[-mA ts]] -> -mA ts

Solve[
      Exp[-((αB t(α^(β + 1)))/(β + 
                  1))] == 
        Exp[-((αA (ts^(β + 1)))/(β + 1))], 
      tα]
tα = tα /. %[[1]] /. 
        Log[Exp[-(((ts^(1 + β)) αA)/(1 + β))]] -> -(((ts^(1 + β)) αA)/(1 + β))

The expected longevity of workers in colonies without polyethism P0 depends on probability of survival sm associated with aging-independent mortality and probability of survival sα associated with aging-related mortality. The probability of survival to age t associated with aging-independent mortality is given by

sm = Exp[-(mA f + mB (1 - f)) t];

The probability of survival to age t associated with aging-related mortality equals the probability of surviving B-type tasks during proportion (1-f) of time t, taking into account the correction term tα, where ts is replaced by the proportion f of time t spent performing the A-type tasks.

sα = 
        Exp[-(1/(β + 
                    1)) αB ((((1 - f) t + ((α(A/α)B)^(1/(β + 1))) f t))^(β + 1))];

The expected longevity of workers in colonies without polyethism is given by

p0x = (∫_0^∞) 
            sm sα \[DifferentialD]t;

Numerical values were assigned to the parameters.

αA = 1.*^-7;
αB = 1.*^-6;
β = 3;
f = 0.5;
mA = 0.01;

Approximate function tsi represents the switching time ts.

tsv[(mB_)?NumberQ]  :=  
      ts  /. FindRoot[(aAx/(aAx + aBx)) == f,  {ts,  20}]
tsi =  
        FunctionInterpolation[tsv[mB], {mB,  0.001,  0.06}];

To simplify computations the expected longevity of workers in colonies with age polyethism is expressed as

pax = (aAx/f);

Switching time ts is replaced with approximate function tsi.

pax = pax /. ts -> tsi[mB];

The difference in expected longevity of workers between colonies with and without age polyethism is plotted.

Plot[pax-p0x,{mB, 0.001, 0.06},AxesLabel->{"mB","pax-p0x"}];