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An oscillatory model revisited
Chaos, Solitons and Fractals 32 (2007) 1325–1327 www.elsevier.com/locate/chaos
An oscillatory model revisited J.H. Swart, H.C. Murrell
*
School of Computer Science, Pietermaritzburg Campus, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, South Africa Accepted 28 November 2005
Abstract A generalised Verhulst population model is considered, wherein the parameters are periodic in time. Some properties of the solutions are discussed. Ó 2005 Published by Elsevier Ltd.
1. Introduction In an interesting paper [2] Leach and Andriopoulos analyze a simple population model with time-dependent parameters, and point out some errors and misinterpretations in a recent work [1] of Lakshmi. Unfortunately they appear to be guilty of some of their own. In [2], Leach and Andriopoulos consider the generalised Verhulst model P_ ¼ rP ðM P Þ the solution of which is indeed easily seen to be R t P 0 exp 0 rðsÞMðsÞ ds R s . P ðtÞ ¼ Rt 1 þ P 0 0 rðsÞ exp 0 rðuÞMðuÞ du ds
ð1Þ
ð2Þ
In [2], Leach and Andriopoulos eloquently argue that if it is assumed that rðtÞ ¼ c sin t; MðtÞ ¼ a þ b cos t;
ð3Þ
where all constants are positive and a > b, then the solutions given in Eq. (2) are not periodic. After some reflection, however, it appears that the solutions given in Eq. (2) are indeed periodic. In fact we will prove a somewhat more general result. Theorem. Under the assumptions that r(t) is an odd function and M(t) is an even function and that both are periodic with period T, all the solutions, P(t) given in Eq. (2), are periodic with period T. Furthermore, if P0 6 M, where M is any lower bound of M(t), then P(t) is positive and bounded.
*
Corresponding author. E-mail address: [email protected] (H.C. Murrell).
0960-0779/$ - see front matter Ó 2005 Published by Elsevier Ltd. doi:10.1016/j.chaos.2005.11.083
1326
J.H. Swart, H.C. Murrell / Chaos, Solitons and Fractals 32 (2007) 1325–1327
Proof. We make use of two trivial results R 0viz RT RT if f(t) is periodic with period T, then T f ðtÞ dt ¼ 0 f ðtÞ dt, and if furthermore f(t) is odd, then 0 f ðtÞ dt ¼ 0. The numerator in Eq. (2) involves Z t IðtÞ rðsÞMðsÞ ds. 0
It is easily shown that I(t) is periodic with period T, and is, in fact, an even function. Now consider the denominator in Eq. (2). If Z s Z t Z t rðsÞ exp rðuÞMðuÞ du ds ¼ rðsÞ exp½IðsÞ ds; QðtÞ ¼ 0
0
0
then Z
Qðt þ T Þ ¼
tþT
rðsÞ exp IðsÞ ds ¼
Z
t
rðu þ T Þ exp Iðu þ T Þ du ¼
T
0
Z
0
rðuÞ exp IðuÞ du þ
Z
T
t
rðuÞ exp IðuÞ du 0
¼ 0 þ QðtÞ. Thus since all terms in Eq. (2) are periodic, it follows that P(t) is periodic. There is still the possibility that P(t) may blow up in (0, T), i.e. the denominator in Eq. (2) may become zero for some value t* in (0, T). We shall show that this cannot happen if P0 6 M. Without loss of generality we assume r(t) increases initially and then oscillates randomly becoming zero at the points ti, i = 1, 2, . . . , n in (0, T). The denominator in Eq. (2) will obviously be positive for small values of t and then may become zero for the first time at some value t*. It is clear that t* can only be in an interval (t2k+1, t2k+2) where r(t) is negative. We illustrate our proof by first considering a simple case. If t* were to be in (t1, t2), we would have Z s Z s Z t1 Z t rðsÞ exp rðuÞMðuÞ du ds þ rðsÞ exp rðuÞMðuÞ du ds Qðt Þ ¼ 0
Z
t1
0
Z
s
rðsÞ exp M
P 0
¼ M 1
Z rðuÞ du þ
0
0
t1
t
Z s rðsÞ exp M rðuÞ du ds 0
t1
Z 1 þ exp M
t
rðuÞ du .
0
Hence the denominator in Eq. (2) satisfies Z 1 þ P 0 QðtÞ P 1 P 0 M 1 þ P 0 M 1 exp M
t
rðuÞ du > 0.
0
t*
if P0 6 M, and cannot be in (t1, t2). This analysis readily generalises to the case where t* may be in some interval (t2k+1, t2k+2) Z t2k Z t2kþ1 Z t rðsÞ exp IðsÞ ds þ rðsÞ exp IðsÞ ds þ rðsÞ exp IðsÞ ds Qðt Þ ¼ 0
t2k
Z 1 P M 1 þ exp M Z þ M 1 exp M
rðsÞ ds
0
t
t2k
t2kþ1
Z 1 þ M exp M
Z rðsÞ ds exp M
0
Z ¼ M 1 1 þ exp M
t
rðsÞ ds .
2kþ1
rðsÞ ds exp M 0
t2kþ1
Z
t2k
rðsÞ ds 0
rðsÞ ds
0
0
R t Hence, again, 1 þ P 0 Qðt Þ P 1 P 0 M 1 þ M 1 exp½M 0 rðsÞ ds > 0 if P0 6 M. Thus no such t* can exist, and P(t) is a positive and bounded periodic function. In Fig. 1 a solution, P(t), is illustrated, with the parameters set as P0 = 1, a = 2, b = 1, and c = 6. It is interesting to note that the authors of [2] tried unsuccessfully to plot P(t) with c set to 10 which probably led them to claim that P(t) was not periodic with this selection of parameters. The reason for the difficulty is the magnitude
J.H. Swart, H.C. Murrell / Chaos, Solitons and Fractals 32 (2007) 1325–1327
1327
2.75 2.5 2.25 P
2 1.75 1.5 1.25 2
4
6
8
10
12
t
Fig. 1. P[t] with parameters P0 = 1, a = 2, b = 1, c = 6.
6 4 2 P 2
4
6
8
10
12
t
-2 -4
Fig. 2. P[t] with parameters P0 = 1, a = 2, b = 1, c = 10.
of the numbers appearing in the numerator and denominator of (2). A standard numerical trick, of dividing the numerator and denominator by the numerator, yields a dramatically changed, and evidently periodic picture shown in Fig. 2. Although the evaluation of P(t) for some values of t is still somewhat unstable the periodic nature of P(t) is clear. This curious change (to where the species comes painfully close to extinction) is probably not for a biological reason, but as a natural consequence of the model. Contrary to the assertion in [2] that the integral in Eq. (2) cannot be evaluated in closed form, the calculation can be done, with a little help from Mathematica. First we employ the transformation cos s = u to rewrite Eq. (2) as P ðtÞ ¼
P 0 exp½acð1 cos tÞ þ 12 bcð1 cos2 tÞ . R1 1 þ cP 0 cos t exp½acð1 uÞ þ 12 bcð1 uÞ2 Þ du
ð4Þ
This form of the solution appears in [2]. A simple completion of the squares and a linear transformation of the form x = [c(2b) 1]1/2(bu + a) leads to the explicit form of the solution P ðtÞ ¼
P 0 exp½acð1 cos tÞ þ 12 bcð1 cos2 tÞ ðaþbÞ2 c pffi hðaþbÞpffici hðaþb cos tÞpffici . pffi pffi pffi c exp P 0 p2 erf pffi pffi erf 2b 2 b 2 b pffiffi 1þ
ð5Þ
b
Because of the presence of error functions, which act rather like Heaviside step functions for a rapidly growing argument, some spikiness will appear locally in the solution.
References [1] Lakshmi BB. Oscillating population models. Chaos, Solitons & Fractals 2003;16:183–6. [2] Andriopoulos K, Leach PGL. An oscillatory population model. Chaos, Solitons & Fractals 2004;22:1183–8.
An oscillatory model revisited J.H. Swart, H.C. Murrell
*
School of Computer Science, Pietermaritzburg Campus, University of KwaZulu-Natal, Private Bag X01, Scottsville 3209, South Africa Accepted 28 November 2005
Abstract A generalised Verhulst population model is considered, wherein the parameters are periodic in time. Some properties of the solutions are discussed. Ó 2005 Published by Elsevier Ltd.
1. Introduction In an interesting paper [2] Leach and Andriopoulos analyze a simple population model with time-dependent parameters, and point out some errors and misinterpretations in a recent work [1] of Lakshmi. Unfortunately they appear to be guilty of some of their own. In [2], Leach and Andriopoulos consider the generalised Verhulst model P_ ¼ rP ðM P Þ the solution of which is indeed easily seen to be R t P 0 exp 0 rðsÞMðsÞ ds R s . P ðtÞ ¼ Rt 1 þ P 0 0 rðsÞ exp 0 rðuÞMðuÞ du ds
ð1Þ
ð2Þ
In [2], Leach and Andriopoulos eloquently argue that if it is assumed that rðtÞ ¼ c sin t; MðtÞ ¼ a þ b cos t;
ð3Þ
where all constants are positive and a > b, then the solutions given in Eq. (2) are not periodic. After some reflection, however, it appears that the solutions given in Eq. (2) are indeed periodic. In fact we will prove a somewhat more general result. Theorem. Under the assumptions that r(t) is an odd function and M(t) is an even function and that both are periodic with period T, all the solutions, P(t) given in Eq. (2), are periodic with period T. Furthermore, if P0 6 M, where M is any lower bound of M(t), then P(t) is positive and bounded.
*
Corresponding author. E-mail address: [email protected] (H.C. Murrell).
0960-0779/$ - see front matter Ó 2005 Published by Elsevier Ltd. doi:10.1016/j.chaos.2005.11.083
1326
J.H. Swart, H.C. Murrell / Chaos, Solitons and Fractals 32 (2007) 1325–1327
Proof. We make use of two trivial results R 0viz RT RT if f(t) is periodic with period T, then T f ðtÞ dt ¼ 0 f ðtÞ dt, and if furthermore f(t) is odd, then 0 f ðtÞ dt ¼ 0. The numerator in Eq. (2) involves Z t IðtÞ rðsÞMðsÞ ds. 0
It is easily shown that I(t) is periodic with period T, and is, in fact, an even function. Now consider the denominator in Eq. (2). If Z s Z t Z t rðsÞ exp rðuÞMðuÞ du ds ¼ rðsÞ exp½IðsÞ ds; QðtÞ ¼ 0
0
0
then Z
Qðt þ T Þ ¼
tþT
rðsÞ exp IðsÞ ds ¼
Z
t
rðu þ T Þ exp Iðu þ T Þ du ¼
T
0
Z
0
rðuÞ exp IðuÞ du þ
Z
T
t
rðuÞ exp IðuÞ du 0
¼ 0 þ QðtÞ. Thus since all terms in Eq. (2) are periodic, it follows that P(t) is periodic. There is still the possibility that P(t) may blow up in (0, T), i.e. the denominator in Eq. (2) may become zero for some value t* in (0, T). We shall show that this cannot happen if P0 6 M. Without loss of generality we assume r(t) increases initially and then oscillates randomly becoming zero at the points ti, i = 1, 2, . . . , n in (0, T). The denominator in Eq. (2) will obviously be positive for small values of t and then may become zero for the first time at some value t*. It is clear that t* can only be in an interval (t2k+1, t2k+2) where r(t) is negative. We illustrate our proof by first considering a simple case. If t* were to be in (t1, t2), we would have Z s Z s Z t1 Z t rðsÞ exp rðuÞMðuÞ du ds þ rðsÞ exp rðuÞMðuÞ du ds Qðt Þ ¼ 0
Z
t1
0
Z
s
rðsÞ exp M
P 0
¼ M 1
Z rðuÞ du þ
0
0
t1
t
Z s rðsÞ exp M rðuÞ du ds 0
t1
Z 1 þ exp M
t
rðuÞ du .
0
Hence the denominator in Eq. (2) satisfies Z 1 þ P 0 QðtÞ P 1 P 0 M 1 þ P 0 M 1 exp M
t
rðuÞ du > 0.
0
t*
if P0 6 M, and cannot be in (t1, t2). This analysis readily generalises to the case where t* may be in some interval (t2k+1, t2k+2) Z t2k Z t2kþ1 Z t rðsÞ exp IðsÞ ds þ rðsÞ exp IðsÞ ds þ rðsÞ exp IðsÞ ds Qðt Þ ¼ 0
t2k
Z 1 P M 1 þ exp M Z þ M 1 exp M
rðsÞ ds
0
t
t2k
t2kþ1
Z 1 þ M exp M
Z rðsÞ ds exp M
0
Z ¼ M 1 1 þ exp M
t
rðsÞ ds .
2kþ1
rðsÞ ds exp M 0
t2kþ1
Z
t2k
rðsÞ ds 0
rðsÞ ds
0
0
R t Hence, again, 1 þ P 0 Qðt Þ P 1 P 0 M 1 þ M 1 exp½M 0 rðsÞ ds > 0 if P0 6 M. Thus no such t* can exist, and P(t) is a positive and bounded periodic function. In Fig. 1 a solution, P(t), is illustrated, with the parameters set as P0 = 1, a = 2, b = 1, and c = 6. It is interesting to note that the authors of [2] tried unsuccessfully to plot P(t) with c set to 10 which probably led them to claim that P(t) was not periodic with this selection of parameters. The reason for the difficulty is the magnitude
J.H. Swart, H.C. Murrell / Chaos, Solitons and Fractals 32 (2007) 1325–1327
1327
2.75 2.5 2.25 P
2 1.75 1.5 1.25 2
4
6
8
10
12
t
Fig. 1. P[t] with parameters P0 = 1, a = 2, b = 1, c = 6.
6 4 2 P 2
4
6
8
10
12
t
-2 -4
Fig. 2. P[t] with parameters P0 = 1, a = 2, b = 1, c = 10.
of the numbers appearing in the numerator and denominator of (2). A standard numerical trick, of dividing the numerator and denominator by the numerator, yields a dramatically changed, and evidently periodic picture shown in Fig. 2. Although the evaluation of P(t) for some values of t is still somewhat unstable the periodic nature of P(t) is clear. This curious change (to where the species comes painfully close to extinction) is probably not for a biological reason, but as a natural consequence of the model. Contrary to the assertion in [2] that the integral in Eq. (2) cannot be evaluated in closed form, the calculation can be done, with a little help from Mathematica. First we employ the transformation cos s = u to rewrite Eq. (2) as P ðtÞ ¼
P 0 exp½acð1 cos tÞ þ 12 bcð1 cos2 tÞ . R1 1 þ cP 0 cos t exp½acð1 uÞ þ 12 bcð1 uÞ2 Þ du
ð4Þ
This form of the solution appears in [2]. A simple completion of the squares and a linear transformation of the form x = [c(2b) 1]1/2(bu + a) leads to the explicit form of the solution P ðtÞ ¼
P 0 exp½acð1 cos tÞ þ 12 bcð1 cos2 tÞ ðaþbÞ2 c pffi hðaþbÞpffici hðaþb cos tÞpffici . pffi pffi pffi c exp P 0 p2 erf pffi pffi erf 2b 2 b 2 b pffiffi 1þ
ð5Þ
b
Because of the presence of error functions, which act rather like Heaviside step functions for a rapidly growing argument, some spikiness will appear locally in the solution.
References [1] Lakshmi BB. Oscillating population models. Chaos, Solitons & Fractals 2003;16:183–6. [2] Andriopoulos K, Leach PGL. An oscillatory population model. Chaos, Solitons & Fractals 2004;22:1183–8.