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The Approximately Ideal, More or Less Free Distribution
Theoretical Population Biology 61, 249–250 (2002) doi:10.1006/tpbi.2001.1560, available online at http://www.idealibrary.com on
CORRIGENDUM J. G. Ollason, N. Ren, and J. M. Yearsley Culterty Field Station, Department of Zoology, University of Aberdeen, Newburgh, Ellon, Aberdeenshire AB41 6AA, Scotland E-mail: [email protected]
Volume 59, Number 2 (2001), in the article ‘‘The Approximately Ideal, More or Less Free Distribution,’’ by J. G. Ollason and J. M. Yearsley, pages 87–105 (doi: 10.1006/tpbi.2000.1505): On page 91, Eq. (12) is invalid. The equation in question purported to show the relationship between the equilibrium staying times of a hungry forager feeding in an environment containing two patches of food regenerating at constant rates. In contrast to the original equation, the corrected equation, following, shows that the ratio of the equilibrium staying times of a forager as a function of the rates of regeneration is dependent on the travelling time, tt , between the patches, as the original equation purported to prove, but is also dependent on n, the specific feeding rate,
1 1
2 2
1 (t0 +2tt ) −nt1 + e a0 n (1 − e −nt1 ) lim = , r Q 0 a1 1 (t1 +2tt ) −nt0 + e n (1 − e −nt0 )
(1)
where a0 and a1 are the rates of regeneration, t0 and t1 are the equilibrium staying times of the animal at Patches 0 and 1, respectively, tt is the travelling time between the patches, and r is the specific metabolic rate. Given that a0 < a1 , m0 (t0 ) < m1 (t1 ), and further analysis confirms the new result lim rQ0
t0 a t +t [ 0 [ 0 t; t1 +2tt a1 t1 +tt
this leads to lim rQ0 tt Q 0
t0 a0 = , t1 a1
which was also asserted in the original analysis, and which is now confirmed to be correct, and to lim rQ0 tt Q .
t0 =1, t1
which was also correctly derived in the original analysis, proving that the degree of undermatching is an increasing function of travelling, time.
⁄
249 © 2002 Elsevier Science (USA) All rights reserved.
250
Corrigendum
Two other limits have been newly derived: lim rQ0 nQ0
t0 a0 = t1 a1
and
lim rQ0 nQ.
t0 =1. t1
These are derived from Eq. (1) and from a further analysis of the equilibrium values of the staying times at each patch. The biological significance of these findings is that the mismatching of the ideal free distribution by hungry foragers is, at least, caused not only by the interaction of the two parameters, r, the specific metabolic rate, and tt , the travelling time, as asserted originally, but also by n, the specific feeding rate. When any of these is increased, the proportion of the total time that is spent feeding is reduced and the proportion of total time spent travelling is correspondingly increased, increasing the degree of undermatching. The results of the simulations described in the paper, and the conclusions drawn from them, are independent of the mathematical error corrected here.
Printed in Belgium
CORRIGENDUM J. G. Ollason, N. Ren, and J. M. Yearsley Culterty Field Station, Department of Zoology, University of Aberdeen, Newburgh, Ellon, Aberdeenshire AB41 6AA, Scotland E-mail: [email protected]
Volume 59, Number 2 (2001), in the article ‘‘The Approximately Ideal, More or Less Free Distribution,’’ by J. G. Ollason and J. M. Yearsley, pages 87–105 (doi: 10.1006/tpbi.2000.1505): On page 91, Eq. (12) is invalid. The equation in question purported to show the relationship between the equilibrium staying times of a hungry forager feeding in an environment containing two patches of food regenerating at constant rates. In contrast to the original equation, the corrected equation, following, shows that the ratio of the equilibrium staying times of a forager as a function of the rates of regeneration is dependent on the travelling time, tt , between the patches, as the original equation purported to prove, but is also dependent on n, the specific feeding rate,
1 1
2 2
1 (t0 +2tt ) −nt1 + e a0 n (1 − e −nt1 ) lim = , r Q 0 a1 1 (t1 +2tt ) −nt0 + e n (1 − e −nt0 )
(1)
where a0 and a1 are the rates of regeneration, t0 and t1 are the equilibrium staying times of the animal at Patches 0 and 1, respectively, tt is the travelling time between the patches, and r is the specific metabolic rate. Given that a0 < a1 , m0 (t0 ) < m1 (t1 ), and further analysis confirms the new result lim rQ0
t0 a t +t [ 0 [ 0 t; t1 +2tt a1 t1 +tt
this leads to lim rQ0 tt Q 0
t0 a0 = , t1 a1
which was also asserted in the original analysis, and which is now confirmed to be correct, and to lim rQ0 tt Q .
t0 =1, t1
which was also correctly derived in the original analysis, proving that the degree of undermatching is an increasing function of travelling, time.
⁄
249 © 2002 Elsevier Science (USA) All rights reserved.
250
Corrigendum
Two other limits have been newly derived: lim rQ0 nQ0
t0 a0 = t1 a1
and
lim rQ0 nQ.
t0 =1. t1
These are derived from Eq. (1) and from a further analysis of the equilibrium values of the staying times at each patch. The biological significance of these findings is that the mismatching of the ideal free distribution by hungry foragers is, at least, caused not only by the interaction of the two parameters, r, the specific metabolic rate, and tt , the travelling time, as asserted originally, but also by n, the specific feeding rate. When any of these is increased, the proportion of the total time that is spent feeding is reduced and the proportion of total time spent travelling is correspondingly increased, increasing the degree of undermatching. The results of the simulations described in the paper, and the conclusions drawn from them, are independent of the mathematical error corrected here.
Printed in Belgium