- Email: [email protected]
Overcrowding effect, unstable or stable
e c o l o g i c a l m o d e l l i n g 2 0 0 ( 2 0 0 7 ) 269–270
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/ecolmodel
Letter to the Editor
Overcrowding effect, unstable or stable
Hui and Li (2003) illustrated that the overcrowding effect was the key factor to incur the chaotic dynamics in the metapopulation model. Besides the overcrowding effect, Allee effect and rescue effect also increase the dynamic complexity. However, through simulating the Eq. (4) of Hui and Li (2003) with the same parameter values, we find that there exist 3-cycles, chaos, limit cycles and stable equilibrium, with the increasing of overcrowding effect intensity (D 40) (shown in Fig. 1). Here we will argue that as the overcrowding effect can also stabilize the system dynamics. Thus, it is unreasonable to say that the overcrowding effect is the key to destabilizing the system dynamics. Let us consider the Eq. (4) of Hui and Li (2003) with h = 1, dP = cp(1 − P) − (e + (1 − e)PD )P dt
(1)
With the condition that D → ∞, the equation becomes the Levins’ (1969) patch occupant model. Studies have shown that the Levins model cannot have any dynamic complexity unless the colonization rate c > 2.57 (May, 1974), which is impossible in metapopulation model (0 < c < 1). So, it is not perfect and
neglect a global cognition, that “because only overcrowding effect can incur chaos, we let it strong enough to kindle chaos at first” (Hui and Li, 2003). First, the definition of parameter D requires clarification. In Hui and Li’s (2003) work, D denotes the intensity of overcrowding effect. They suggested that the increase of overcrowding effect intensity may result in chaos, which means that the more overcrowded the metapopulation is, the more complex its dynamics are. Actually, this conclusion is misguided due to the incorrect definition of parameter D. In fact, the overcrowding effect should increase the extinction rate of the metapopulation. However, the extinction rate (e + (1 − e)PD ) is actually decreasing with the increasing of parameter D (D > 0). So it is more accurate to consider the large value of parameter D as the weaker overcrowding intensity. Then, in the case where D → 0 (the highest overcrowding intensity), the term (e + (1 − e)PD ) → 1, indicating that the metapopulation is entirely extinct. This case is reflected at the beginning phase of bifurcation diagrams (Fig. 1). When D → ∞ (representing that the metapopulation is completely un-effected by the overcrowding effect), the term (e + (1 − e)PD ) → e, which indicates that the metapopulation will persist as long as c > e (Hanski, 1999). This is the reason the dynamics of the system stabilizes when the parameter D is large. Thus, it will be more appropriate to use d (d = 1/D), instead of D, as the intensity of the overcrowding effect. With this description, the Eq. (1) becomes, dP = cp(1 − P) − (e + (1 − e)P1/d )P dt
Fig. 1 – Bifurcation diagrams of metapopulation dynamics with overcrowding effect. Parameter values are c = 0.99, e = 0.01, h = 1.
(2)
In Eq. (2), the extinction rate (e + (1 − e)P1/d is an increasing function of parameter d. We may draw the conclusion that as the overcrowding intensity increases, the metapopulation will lose its stable persistent state and become extinct rapidly. When the local abundance or density of a local population is too large and it stirs up the overcrowding effect, the extinction rate of local population will rapidly increase. So far, the explicit theoretical and empirical investigations of the overcrowding effect are still scarce (but see Buffoni and Gilioli, 2003; Hui and Li, 2003; Hui, 2004). Objectively speaking, Hui and Li’s work is the first to propose the Levins model sub-
270
e c o l o g i c a l m o d e l l i n g 2 0 0 ( 2 0 0 7 ) 269–270
jected to overcrowding effect and discuss the influences of overcrowding effect on dynamic complexity and the spatial distribution of metapopulation, although the conclusions they drew are inadequate (e.g. ambiguous definition of overcrowding intensity and its value range). Here, we offer the valuable supplement of further analysis of their equation and clarifying the implication of parameters. It’s important to realize the exact meaning of overcrowding effect.
Levins, R., 1969. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entomol. Soc. Am. 15, 237–240. May, R.M., 1974. Biological populations with non-overlapping generations: stable points, stable cycles and chaos. Science 186, 645–647.
Yanyu Zhang Key Laboratory of Arid Agroecology under the Ministry of Education, School of Life Science, Lanzhou University, Lanzhou 730000, China
Acknowledgements We are grateful to J. Jia and M. Su for their constructive comments and kind help to with the English of this manuscript. We also thank the help comments received from the anonymous reviews and the editors. This work was supported by the National Natural Science Foundation of China (No. 30470298) and the National Social Science Foundation of China (No. 04AJL007).
references
Buffoni, G., Gilioli, G., 2003. A lumped parameter model for acarine predator-prey population interactions. Ecol. Model. 170, 155–171. Hanski, I., 1999. Metapopulation Ecology. Oxford University Press, Oxford. Hui, C., 2004. Spatial chaos of metapopulation incurred by Allee effect, overcrowding effect and predation effect. Acta Bot. Boreal. Occident. Sin. 24, 370–383. Hui, C., Li, Z., 2003. Dynamical complexity and metapopulation persistence. Ecol. Model. 164, 201–209.
Xiaozhuo Han Faculty of Applied Mathematics, Guangdong University of Technology, Guangzhou 510090, China a
b
Zizhen Li a,b,∗ Key Laboratory of Arid Agroecology under the Ministry of Education, School of Life Science, Lanzhou University, Lanzhou 730000, China
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
∗ Corresponding
author at: Key Laboratory of Arid Agroecology under the Ministry of Education, School of Life Science, Lanzhou University, Lanzhou 730000, China. Tel.: +86 931 8913370; fax: +86 931 8912823.
E-mail addresses: [email protected] (Y. Zhang), [email protected] (X. Han), [email protected] (Z. Li) Published on line 19 October 2006 0304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2006.09.007
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/ecolmodel
Letter to the Editor
Overcrowding effect, unstable or stable
Hui and Li (2003) illustrated that the overcrowding effect was the key factor to incur the chaotic dynamics in the metapopulation model. Besides the overcrowding effect, Allee effect and rescue effect also increase the dynamic complexity. However, through simulating the Eq. (4) of Hui and Li (2003) with the same parameter values, we find that there exist 3-cycles, chaos, limit cycles and stable equilibrium, with the increasing of overcrowding effect intensity (D 40) (shown in Fig. 1). Here we will argue that as the overcrowding effect can also stabilize the system dynamics. Thus, it is unreasonable to say that the overcrowding effect is the key to destabilizing the system dynamics. Let us consider the Eq. (4) of Hui and Li (2003) with h = 1, dP = cp(1 − P) − (e + (1 − e)PD )P dt
(1)
With the condition that D → ∞, the equation becomes the Levins’ (1969) patch occupant model. Studies have shown that the Levins model cannot have any dynamic complexity unless the colonization rate c > 2.57 (May, 1974), which is impossible in metapopulation model (0 < c < 1). So, it is not perfect and
neglect a global cognition, that “because only overcrowding effect can incur chaos, we let it strong enough to kindle chaos at first” (Hui and Li, 2003). First, the definition of parameter D requires clarification. In Hui and Li’s (2003) work, D denotes the intensity of overcrowding effect. They suggested that the increase of overcrowding effect intensity may result in chaos, which means that the more overcrowded the metapopulation is, the more complex its dynamics are. Actually, this conclusion is misguided due to the incorrect definition of parameter D. In fact, the overcrowding effect should increase the extinction rate of the metapopulation. However, the extinction rate (e + (1 − e)PD ) is actually decreasing with the increasing of parameter D (D > 0). So it is more accurate to consider the large value of parameter D as the weaker overcrowding intensity. Then, in the case where D → 0 (the highest overcrowding intensity), the term (e + (1 − e)PD ) → 1, indicating that the metapopulation is entirely extinct. This case is reflected at the beginning phase of bifurcation diagrams (Fig. 1). When D → ∞ (representing that the metapopulation is completely un-effected by the overcrowding effect), the term (e + (1 − e)PD ) → e, which indicates that the metapopulation will persist as long as c > e (Hanski, 1999). This is the reason the dynamics of the system stabilizes when the parameter D is large. Thus, it will be more appropriate to use d (d = 1/D), instead of D, as the intensity of the overcrowding effect. With this description, the Eq. (1) becomes, dP = cp(1 − P) − (e + (1 − e)P1/d )P dt
Fig. 1 – Bifurcation diagrams of metapopulation dynamics with overcrowding effect. Parameter values are c = 0.99, e = 0.01, h = 1.
(2)
In Eq. (2), the extinction rate (e + (1 − e)P1/d is an increasing function of parameter d. We may draw the conclusion that as the overcrowding intensity increases, the metapopulation will lose its stable persistent state and become extinct rapidly. When the local abundance or density of a local population is too large and it stirs up the overcrowding effect, the extinction rate of local population will rapidly increase. So far, the explicit theoretical and empirical investigations of the overcrowding effect are still scarce (but see Buffoni and Gilioli, 2003; Hui and Li, 2003; Hui, 2004). Objectively speaking, Hui and Li’s work is the first to propose the Levins model sub-
270
e c o l o g i c a l m o d e l l i n g 2 0 0 ( 2 0 0 7 ) 269–270
jected to overcrowding effect and discuss the influences of overcrowding effect on dynamic complexity and the spatial distribution of metapopulation, although the conclusions they drew are inadequate (e.g. ambiguous definition of overcrowding intensity and its value range). Here, we offer the valuable supplement of further analysis of their equation and clarifying the implication of parameters. It’s important to realize the exact meaning of overcrowding effect.
Levins, R., 1969. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entomol. Soc. Am. 15, 237–240. May, R.M., 1974. Biological populations with non-overlapping generations: stable points, stable cycles and chaos. Science 186, 645–647.
Yanyu Zhang Key Laboratory of Arid Agroecology under the Ministry of Education, School of Life Science, Lanzhou University, Lanzhou 730000, China
Acknowledgements We are grateful to J. Jia and M. Su for their constructive comments and kind help to with the English of this manuscript. We also thank the help comments received from the anonymous reviews and the editors. This work was supported by the National Natural Science Foundation of China (No. 30470298) and the National Social Science Foundation of China (No. 04AJL007).
references
Buffoni, G., Gilioli, G., 2003. A lumped parameter model for acarine predator-prey population interactions. Ecol. Model. 170, 155–171. Hanski, I., 1999. Metapopulation Ecology. Oxford University Press, Oxford. Hui, C., 2004. Spatial chaos of metapopulation incurred by Allee effect, overcrowding effect and predation effect. Acta Bot. Boreal. Occident. Sin. 24, 370–383. Hui, C., Li, Z., 2003. Dynamical complexity and metapopulation persistence. Ecol. Model. 164, 201–209.
Xiaozhuo Han Faculty of Applied Mathematics, Guangdong University of Technology, Guangzhou 510090, China a
b
Zizhen Li a,b,∗ Key Laboratory of Arid Agroecology under the Ministry of Education, School of Life Science, Lanzhou University, Lanzhou 730000, China
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
∗ Corresponding
author at: Key Laboratory of Arid Agroecology under the Ministry of Education, School of Life Science, Lanzhou University, Lanzhou 730000, China. Tel.: +86 931 8913370; fax: +86 931 8912823.
E-mail addresses: [email protected] (Y. Zhang), [email protected] (X. Han), [email protected] (Z. Li) Published on line 19 October 2006 0304-3800/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolmodel.2006.09.007