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Comments to the effect of prey refuge in a simple predator–prey model
Ecological Modelling 287 (2014) 58–59
Contents lists available at ScienceDirect
Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel
Letter to the Editor Comments to the effect of prey refuge in a simple predator–prey model夽
On the other hand, the inequation S < 0 is equivalent to the following inequations:
The predator–prey model with the effect of prey refuge was studied by many scholar (Hassell and May, 1973; Hoy, 1985; Krivan, 1998; González-Olivares and Ramos-Jiliberto, 2003; Kar, 2005; Huang et al., 2006; Ma et al., 2009). Some of the empirical and theoretical work drawn a conclusion that the refuges used by prey have a stabilizing effect on the considered interactions and prey extinction can be prevented by the addition of refuges (Krivan, 1998; Kar, 2005; Huang et al., 2006; Ma et al., 2009, 2011, 2013; Ma and Wang, 2012). González-Olivares and Ramos-Jiliberto (2003) considered the following predator–prey model with prey refuges
K p ac + − − 2 ac p−c
⎧ dx x q(x − xr )y ⎪ = r 1 − x− , ⎨ dt K x − xr + a ⎪ ⎩ dy = b p(x − xr )y − c y. dt
(1)
x − xr + a
They studied the dynamic consequences of prey refuges in a simple model and obtained that the community equilibrium being stabilized by the addition of refuges for prey population, and prey extinction being prevented. According to their analysis about the second case while a constant number of prey use refuges (xr = ), the following conclusions are obtained (Theorem 4.3a in GonzálezOlivares and Ramos-Jiliberto (2003)): 1. If S > 0, the singularity (xe , ye ) of the considered system is an unstable equilibrium point, surrounded by a limit cycle. 2. If S < 0, the singularity (xe , ye ) of the considered system is a locally asymptotically stable equilibrium point, where S = (p − c)3 2 − (p − c)(2ac2 + K(p − c)2 ) + ac2 (K(p − c) − a(c − p)). After this work, Ma et al. (2011) tried to modify some computation in the paper studied in González-Olivares and Ramos-Jiliberto (2003) in order to explain the effect of prey refuge explicitly. In their paper, the second case while a constant number of prey use refuges (xr = ) was modified and they claimed that the equation S = (p − c)3 2 − (p − c)(2ac2 + K(p − c)2 ) + ac2 (K(p − c) − a(c − p)) > 0 is equivalent to the following inequations:
K p ac + > + − 2 ac p−c or p ac K < + − − 2 ac p−c
K 2 2
K 2 2
+
+
p 2
p 2 ac
.
夽 This work was supported by the National Natural Science Foundation of China (No. 11301238) and the Fundamental Research Funds for the Central Universities (Nos. 31920930002 and XBMUYJRC1201202). http://dx.doi.org/10.1016/j.ecolmodel.2014.04.027 0304-3800/© 2014 Elsevier B.V. All rights reserved.
+
2
ac
2 K p 2
ac K p − << + + 2 ac p−c
2
+
ac
.
According to the above analysis, they obtained the following results: 1. The interior equilibrium point is locally asymptotically sta2
ble if (K/2) + (p/ac) − (ac/(p − c)) − (K/2) + (p/ac) − (ac/(p − c)) +
2
(K/2) + (p/ac) < <
2
2
(K/2) + (p/ac) , and unstable
2
2
if > (K/2) + (p/ac) − (ac/(p − c)) + (K/2) + (p/ac) . 2. The singularity of the considered model has no ecological meanings while > (K/2) + (p/ac) − (ac/(p − c)) + 2
2
(K/2) + (p/ac) since the interior equilibrium point reduced to the predator-free equilibrium point when the condition > K − (ac/(p − c)) is satisfied. This made Theorem 4.3a (in González-Olivares and Ramos-Jiliberto, 2003) forfeit the ecological meanings in their work. In this sight, Ma et al. think that it is necessary to correct the their results in order to have ecological meanings and find the stabilizing effect clearly.
Unfortunately, Ma et al. (2011) had not achieved their aim since the computation had some mistakes. After that, González-Olivares and Ramos-Jiliberto (2012) found these mistakes and given the correct forms. In fact, S = (p − c)3 2 − (p − c)(2ac2 + K(p − c)2 ) + ac2 (K(p − c) − a(c − p)) = 0 is equivalent to the following inequations
K 2ac 2 = ± + 2 (p − c)2
K 2 2
+
a2 c 2 p(2c − p) (p − c)2
On the other hand, if = a, according to Theorem 5a (GonzálezOlivares and Ramos-Jiliberto, 2003), we can obtain the following Proposition 1 by simple computations. Therefore, according to the above analysis, the correct results are proposed as follows. If = a, we obtain that Proposition 1.
ac
2 K p 2
Supposing c < p < 2c, we have
1. If 0 < < K(1 − (c/p))(1 − (c/(3c − p))), then the prey and predator populations oscillate periodically around the unique equilibrium point at the first quadrant. 2. If K(1 − (c/p))(1 − (c/(3c − p))) < < K(1 − (c/p)), then the two populations tend to reach a globally asymptotically stable equilibrium point. 3. If > K(1 − (c/p)), the prey population tends to reach its maximum value K and the predators are depleted.
Letter to the Editor / Ecological Modelling 287 (2014) 58–59
If = / a, we obtain that Proposition 2. 1. If
References
Supposing c < p < 2c, we have 2
0 < < (K/2) + (2ac 2 /((p − c) )) − 2
(K/2) − p))/((p − c)2 )), then the prey and predator populations oscillate periodically around the unique equilibrium point at the first quadrant. + ((a2 c 2 p(2c
2
2
2
(K/2) + (2ac 2 /((p − c) )) − (K/2) + ((a2 c 2 p(2c − p))/((p − c) )) < < K − (ac/(p − c)), then the two populations tend to reach a
2. If
59
globally asymptotically stable equilibrium point. 3. If > K − (ac/(p − c)), the prey population tends to reach its maximum value K and the predators are depleted. According to the above results, the effect of prey refuges could increase the local stability of the community equilibrium in the first quadrant or had a stabilizing effect for the considered predator–prey model. Here, increase of stability or stabilizing effect refers to cases where a positive equilibrium point changes from repeller to an attractor due to changes in the value of the parameter . When the controlling parameter increases and exceeds the value K − (ac/(p − c)), the results show that the prey population tends to reach its maximum value K and the predators are depleted. This occurs at high levels of refuge used by prey population. Therefore, three kind of state of equilibrium can be reached: unstable point, stable coexistence of two interacting populations, and predator extinction while the prey reaches its carrying capacity. On the other hand, A stable equilibrium can never transform to unstable by the addition or by increasing of constant number refuge, in close agreement with earlier results (González-Olivares and Ramos-Jiliberto, 2003; Kar, 2005; Huang et al., 2006) and disagreement with Ma et al. (2009). They considered a generalized predator–prey model incorporating the effect of prey refuges and found that a stable equilibrium could transform to unstable by increasing of constant proportion refuge. Therefore, the parameter conditions under which each of these equilibrium are expected depend strongly on the mode of refuge applying and predation of predator to prey population.
González-Olivares, E., Ramos-Jiliberto, R., 2012. Comments to “The effect of prey refuge in a simple predator–prey model. [Ecol. Model. 222 (2011) 3453–3454]”. Ecol. Model. 232, 158–160. González-Olivares, E., Ramos-Jiliberto, R., 2003. Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability. Ecol. Model. 166, 135–146. Hassell, M.P., May, R.M., 1973. Stability in insect host–parasite models. J. Anim. Ecol. 42, 693–698. Hoy, M.A., 1985. Almonds (California). In: Helle, W., Sabelis, M.W. (Eds.), In: Spider Mites: Their Biology, Natural Enemies and Control, World Crop Pests, vol. 1B. Elsevier, Amsterdam. Huang, Y., Chen, F., Zhong, L., 2006. Stability analysis of a prey–predator model with Holling type III response function incorporating a prey refuge. Appl. Math. Comput. 182, 672–678. Kar, T.K., 2005. Stability analysis of a prey–predator model incorporating a prey refuge. Commun. Nonlinear Sci. Numer. Simul. 10, 681–691. Krivan, V., 1998. Effects of optimal antipredator behavior of prey on predator–prey dynamics: the role of refuges. Theor. Popul. Biol. 53, 131–142. Ma, Z., Li, W., Zhao, Y., Wang, W., Zhang, H., Li, Z., 2009. Effects of prey refuges on a predator–prey model with a class of functional responses: the role of refuges. Math. Biosci. 218, 73–79. Ma, Z., Li, W., Wang, W., 2011. The effect of prey refuge in a simple predator–prey model. Ecol. Model. 222, 3453–3454. Ma, Z., Wang, W., 2012. Stability analysis: the effect of prey refuge in predator–prey model. Ecol. Model. 247, 95–97. Ma, Z., Wang, W., Li, W., Li, Z., 2013. The effect of prey refuge in a patchy predator–prey system. Math. Biosci. 243, 126–130.
Shufan Wang School of Mathematics and Computer Science Institute, Northwest University for Nationalities, Lanzhou 730030, China Zhihui Ma ∗ School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China ∗ Corresponding
author. Tel.: +86 931 8912483; fax: +86 931 8912823. E-mail address: [email protected] (Z. Ma) 7 March 2014 Available online 13 June 2014
Contents lists available at ScienceDirect
Ecological Modelling journal homepage: www.elsevier.com/locate/ecolmodel
Letter to the Editor Comments to the effect of prey refuge in a simple predator–prey model夽
On the other hand, the inequation S < 0 is equivalent to the following inequations:
The predator–prey model with the effect of prey refuge was studied by many scholar (Hassell and May, 1973; Hoy, 1985; Krivan, 1998; González-Olivares and Ramos-Jiliberto, 2003; Kar, 2005; Huang et al., 2006; Ma et al., 2009). Some of the empirical and theoretical work drawn a conclusion that the refuges used by prey have a stabilizing effect on the considered interactions and prey extinction can be prevented by the addition of refuges (Krivan, 1998; Kar, 2005; Huang et al., 2006; Ma et al., 2009, 2011, 2013; Ma and Wang, 2012). González-Olivares and Ramos-Jiliberto (2003) considered the following predator–prey model with prey refuges
K p ac + − − 2 ac p−c
⎧ dx x q(x − xr )y ⎪ = r 1 − x− , ⎨ dt K x − xr + a ⎪ ⎩ dy = b p(x − xr )y − c y. dt
(1)
x − xr + a
They studied the dynamic consequences of prey refuges in a simple model and obtained that the community equilibrium being stabilized by the addition of refuges for prey population, and prey extinction being prevented. According to their analysis about the second case while a constant number of prey use refuges (xr = ), the following conclusions are obtained (Theorem 4.3a in GonzálezOlivares and Ramos-Jiliberto (2003)): 1. If S > 0, the singularity (xe , ye ) of the considered system is an unstable equilibrium point, surrounded by a limit cycle. 2. If S < 0, the singularity (xe , ye ) of the considered system is a locally asymptotically stable equilibrium point, where S = (p − c)3 2 − (p − c)(2ac2 + K(p − c)2 ) + ac2 (K(p − c) − a(c − p)). After this work, Ma et al. (2011) tried to modify some computation in the paper studied in González-Olivares and Ramos-Jiliberto (2003) in order to explain the effect of prey refuge explicitly. In their paper, the second case while a constant number of prey use refuges (xr = ) was modified and they claimed that the equation S = (p − c)3 2 − (p − c)(2ac2 + K(p − c)2 ) + ac2 (K(p − c) − a(c − p)) > 0 is equivalent to the following inequations:
K p ac + > + − 2 ac p−c or p ac K < + − − 2 ac p−c
K 2 2
K 2 2
+
+
p 2
p 2 ac
.
夽 This work was supported by the National Natural Science Foundation of China (No. 11301238) and the Fundamental Research Funds for the Central Universities (Nos. 31920930002 and XBMUYJRC1201202). http://dx.doi.org/10.1016/j.ecolmodel.2014.04.027 0304-3800/© 2014 Elsevier B.V. All rights reserved.
+
2
ac
2 K p 2
ac K p − << + + 2 ac p−c
2
+
ac
.
According to the above analysis, they obtained the following results: 1. The interior equilibrium point is locally asymptotically sta2
ble if (K/2) + (p/ac) − (ac/(p − c)) − (K/2) + (p/ac) − (ac/(p − c)) +
2
(K/2) + (p/ac) < <
2
2
(K/2) + (p/ac) , and unstable
2
2
if > (K/2) + (p/ac) − (ac/(p − c)) + (K/2) + (p/ac) . 2. The singularity of the considered model has no ecological meanings while > (K/2) + (p/ac) − (ac/(p − c)) + 2
2
(K/2) + (p/ac) since the interior equilibrium point reduced to the predator-free equilibrium point when the condition > K − (ac/(p − c)) is satisfied. This made Theorem 4.3a (in González-Olivares and Ramos-Jiliberto, 2003) forfeit the ecological meanings in their work. In this sight, Ma et al. think that it is necessary to correct the their results in order to have ecological meanings and find the stabilizing effect clearly.
Unfortunately, Ma et al. (2011) had not achieved their aim since the computation had some mistakes. After that, González-Olivares and Ramos-Jiliberto (2012) found these mistakes and given the correct forms. In fact, S = (p − c)3 2 − (p − c)(2ac2 + K(p − c)2 ) + ac2 (K(p − c) − a(c − p)) = 0 is equivalent to the following inequations
K 2ac 2 = ± + 2 (p − c)2
K 2 2
+
a2 c 2 p(2c − p) (p − c)2
On the other hand, if = a, according to Theorem 5a (GonzálezOlivares and Ramos-Jiliberto, 2003), we can obtain the following Proposition 1 by simple computations. Therefore, according to the above analysis, the correct results are proposed as follows. If = a, we obtain that Proposition 1.
ac
2 K p 2
Supposing c < p < 2c, we have
1. If 0 < < K(1 − (c/p))(1 − (c/(3c − p))), then the prey and predator populations oscillate periodically around the unique equilibrium point at the first quadrant. 2. If K(1 − (c/p))(1 − (c/(3c − p))) < < K(1 − (c/p)), then the two populations tend to reach a globally asymptotically stable equilibrium point. 3. If > K(1 − (c/p)), the prey population tends to reach its maximum value K and the predators are depleted.
Letter to the Editor / Ecological Modelling 287 (2014) 58–59
If = / a, we obtain that Proposition 2. 1. If
References
Supposing c < p < 2c, we have 2
0 < < (K/2) + (2ac 2 /((p − c) )) − 2
(K/2) − p))/((p − c)2 )), then the prey and predator populations oscillate periodically around the unique equilibrium point at the first quadrant. + ((a2 c 2 p(2c
2
2
2
(K/2) + (2ac 2 /((p − c) )) − (K/2) + ((a2 c 2 p(2c − p))/((p − c) )) < < K − (ac/(p − c)), then the two populations tend to reach a
2. If
59
globally asymptotically stable equilibrium point. 3. If > K − (ac/(p − c)), the prey population tends to reach its maximum value K and the predators are depleted. According to the above results, the effect of prey refuges could increase the local stability of the community equilibrium in the first quadrant or had a stabilizing effect for the considered predator–prey model. Here, increase of stability or stabilizing effect refers to cases where a positive equilibrium point changes from repeller to an attractor due to changes in the value of the parameter . When the controlling parameter increases and exceeds the value K − (ac/(p − c)), the results show that the prey population tends to reach its maximum value K and the predators are depleted. This occurs at high levels of refuge used by prey population. Therefore, three kind of state of equilibrium can be reached: unstable point, stable coexistence of two interacting populations, and predator extinction while the prey reaches its carrying capacity. On the other hand, A stable equilibrium can never transform to unstable by the addition or by increasing of constant number refuge, in close agreement with earlier results (González-Olivares and Ramos-Jiliberto, 2003; Kar, 2005; Huang et al., 2006) and disagreement with Ma et al. (2009). They considered a generalized predator–prey model incorporating the effect of prey refuges and found that a stable equilibrium could transform to unstable by increasing of constant proportion refuge. Therefore, the parameter conditions under which each of these equilibrium are expected depend strongly on the mode of refuge applying and predation of predator to prey population.
González-Olivares, E., Ramos-Jiliberto, R., 2012. Comments to “The effect of prey refuge in a simple predator–prey model. [Ecol. Model. 222 (2011) 3453–3454]”. Ecol. Model. 232, 158–160. González-Olivares, E., Ramos-Jiliberto, R., 2003. Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability. Ecol. Model. 166, 135–146. Hassell, M.P., May, R.M., 1973. Stability in insect host–parasite models. J. Anim. Ecol. 42, 693–698. Hoy, M.A., 1985. Almonds (California). In: Helle, W., Sabelis, M.W. (Eds.), In: Spider Mites: Their Biology, Natural Enemies and Control, World Crop Pests, vol. 1B. Elsevier, Amsterdam. Huang, Y., Chen, F., Zhong, L., 2006. Stability analysis of a prey–predator model with Holling type III response function incorporating a prey refuge. Appl. Math. Comput. 182, 672–678. Kar, T.K., 2005. Stability analysis of a prey–predator model incorporating a prey refuge. Commun. Nonlinear Sci. Numer. Simul. 10, 681–691. Krivan, V., 1998. Effects of optimal antipredator behavior of prey on predator–prey dynamics: the role of refuges. Theor. Popul. Biol. 53, 131–142. Ma, Z., Li, W., Zhao, Y., Wang, W., Zhang, H., Li, Z., 2009. Effects of prey refuges on a predator–prey model with a class of functional responses: the role of refuges. Math. Biosci. 218, 73–79. Ma, Z., Li, W., Wang, W., 2011. The effect of prey refuge in a simple predator–prey model. Ecol. Model. 222, 3453–3454. Ma, Z., Wang, W., 2012. Stability analysis: the effect of prey refuge in predator–prey model. Ecol. Model. 247, 95–97. Ma, Z., Wang, W., Li, W., Li, Z., 2013. The effect of prey refuge in a patchy predator–prey system. Math. Biosci. 243, 126–130.
Shufan Wang School of Mathematics and Computer Science Institute, Northwest University for Nationalities, Lanzhou 730030, China Zhihui Ma ∗ School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China ∗ Corresponding
author. Tel.: +86 931 8912483; fax: +86 931 8912823. E-mail address: [email protected] (Z. Ma) 7 March 2014 Available online 13 June 2014