Stability of Ecosystem Induced by Mutual Interference between Predators

Available online at www.sciencedirect.com

Procedia Environmental Sciences 2 (2010) 42–48

International Society for Environmental Information Sciences 2010 Annual Conference (ISEIS)

Stability of Ecosystem Induced by Mutual Interference between Predators Jianfeng Fenga*, Lin Zhua, Hongli Wangb a

Key Laboratory of Pollution Processes and Environmental Criteria at Ministry of Education, College of Environmental Science and Engineering, Nankai University, Tianjin 300071, China b Department of Mechanics, School of Mechanical Engineering, Tianjin University, Tianjin 300072, China

Abstract A three-species predator-prey model with Beddington-DeAngelis functional response was established with a food chain including two species of zooplanktons and one of phytoplankton. With nonlinear dynamical theory, the influences of competitions between two zooplanktons and semi-saturation constant on the stability of the system were investigated. Results showed that, in a certain range, increasing competition and saturation constant factor ccould stabilize the ecosystem. We also found that the system only appears Hopf and transcritical bifurcation in a certain range of parameters. Furthermore, the ecological significance of the two bifurcations was presented.

© 2010 Published by Elsevier Ltd. Keywords: Beddington-DeAngelis; Functional response; Mutual interference; Stabilizing influence; Population dynamics

1. Introduction Early studies of the ecological model have focused on the predation role, but neglected the role of mutual interference among predators. In 1975, the Beddington-DeAngelis functional response for the predator-prey model was proposed[1-2]. This function was more fit with the actual natural ecosystems due to the added competition among predators[3-4]. A number of studies have investigated the effect of mutual interference on population dynamics and population stability[6-8]. This model has been used to study the persistence or extinction of species in spatially explicit reaction–diffusion models[9]. From theoretical and empirical studies, a consensus has emerged that interference has a stabilizing influence on population dynamics[10-15], although there is an upper limit on the interference constant beyond which the dynamics becomes unstable[6]. Furthermore, the predator–prey model predicts that increasing

* Corresponding author. Tel.: +86-22-23504379 E-mail address: [email protected].

1878-0296 © 2010 Published by Elsevier doi:10.1016/j.proenv.2010.10.007

Jianfeng Feng et al. / Procedia Environmental Sciences 2 (2010) 42–48

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interference has a positive effect on asymptotic stability and resilience of the biological system when the interference is low, and that the opposite is true when the interference is high[16]. In this paper, we established a predator-prey model involving three species with Beddington-DeAngelis functional response according to a food chain including two species of zooplanktons and one of phytoplankton. We investigated the influence of mutual interference of predators at different trophic levels on the stability of the food– prey–predator chain system. To our knowledge, there have been few studies paying attention to the effect of mutual interference among two predators. Here, we shall demonstrate that in a certain range, the additional mutual interference and saturation constant clearly exerts a stabilizing influence. 2. Model To investigate the effect of mutual interference of predators, according to a food chain including two species of zooplanktons and one of phytoplankton[17], we consider the following model: ­ dX ° dT ° ° dY ® ° dT ° dZ ° dT ¯

where F (U ,V ) i

AU i Bi  U  M iV

RX (1 

X )  F1 ( X , Y )Y  F2 ( X , Z ) Z K

˄1˅

C1F1 ( X , Y )Y  D1Y C2 F2 ( X , Z ) Z  D2 Z

for i 1, 2

In model (1), X denotes the population density of prey, and Y, Z are the population density of two predators. The prey species grow logistically with a maximum growth rate R and carrying capacities K. The trophic function between prey species and predator species was described by a Beddington–DeAngelis functional response with a maximum grazing rate Ai and fixed half saturation value Ai. The factor Mi denotes mutual interference of predators. The parameter Ci is the conversion ratios of prey to predator. The factor Di is the death ratios of species Y and Z respectively. All parameters are positive constants. The model (1) has 12 parameters in all, which make mathematical analysis complex. In order to avoid such difficulties, we reduce the number of parameters by using the following transformations: t

RT , x

X ,y K

AY A Z i , z= 2 RK RK

These substitutions and simplifying yield the following system of equations, where the variables x, y, z are new scaled (nondimensional) measures of population size, and t is a new variable of time:

­ dx xy xz  x(1  x)  ° b1  x  m1 y b2  x  m2 z ° dt ° dy xy  d1 y g2 ( x, y, z) y a1 ® b1  x  m1 y ° dt ° dz xz  d2 z g3 ( x, y, z) z a2 ° b2  x  m2 z ¯ dt

g1 ( x, y, z) x (2)

3. Stability Analysis In this section, the existence and local stability of non-negative equilibrium points in model (2) are investigated. Five non-negative equilibrium points are found in all(A1, A2, A3, A4, A5), we can analyze the local stability of model (2) using its Jacobin matrix at (x, y, z):

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Jianfeng Feng et al. / Procedia Environmental Sciences 2 (2010) 42–48

§ wg1  g1 ¨x ¨ wx ¨ wg 2 ¨ y wx ¨ ¨ wg 3 ¨ z wx ©

J

3.1. A1

§1

0

¨0 ©

0

(0, 0, 0) is : J ¨ 0 d 1 1 ¨

equation of J 2 are: O1

1, O2

(1, 0, 0) is J2

§ 1 ¨ 1 1  b1 ¨ ¨ a1  d1 (b1  1) ¨0 1  b1 ¨ ¨ 0 ¨0 ©

a1  d1 (b1  1) , O3 1  b1

­

(1, 0, 0) is locally and asymptotically stable if : °°

(0, 0, 0) is saddle point.

· ¸ ¸ ¸ ,The roots of the characteristic 0 ¸ ¸ a2  d 2 (b2  1) ¸ ¸ 1  b2 ¹ 1 1  b2

a2  d 2 (b2  1) . According to the Routh-Hurwitz criterion, 1  b2

O2

® °O °¯ 3

3.3. A3

0 · ¸ ˈwhere 1 ! 0 , so A1 0 ¸  d 2 ¸¹

(1, 0, 0)

The Jacobin matrix of A2

A2

x

(0, 0, 0)

The Jacobin matrix of A1

3.2. A2

wg1 · ¸ wz ¸ wg ¸ y 2 ¸ wz ¸ ¸ wg3  g3 ¸ z wz ¹

wg1 wy wg y 2  g2 wy wg z 3 wy x

a1  d1 (b1  1) 0 . 1  b1 a2  d 2 (b2  1) 0 1  b2

( x , y , 0)

The roots of the characteristic equation of the Jacobin matrix of model (2) at A3

( x , y , 0) equilibrium point

satisfy:

xy (1  a1m1 ) ­ °O1  O2  x  ( x  b  ym ) 2 1 1 ° 2 ° x ya1m1 xya1b1  ( x  b1  ym1 ) ®O1O2 2    x b ym x b1  ym1 ) 4 ( ) ( 1 1 ° ° x  d2 °O3 b2  x ¯

z

The stability of A3 ( x , y , 0) in the positive z direction depends on O3 . A3 ( x , y , 0) is stable in the positive direction if . So the condition of stability O3  0

is: x (a1  d1 ) ! b1d1 , G1

(G11  G22 ) ! 0, G 2

G11G22  G12G21 ! 0, O3

x / (b2  x )  d 2  0 .

Jianfeng Feng et al. / Procedia Environmental Sciences 2 (2010) 42–48

3.4. A4

45

( x , 0, z )

The method of mathematical analysis of A4 stability is:

x (a2  d 2 ) ! b2 d 2 , G1

(G11  G22 ) ! 0, G 2

( x , 0, z ) is the same as A3 ( x , y , 0) , and the conditions of

G11G22  G12G21 ! 0, O3

x  d1  0 b1  x

­ w2  w2 2  4a2b2 d 2 m2 ° x The equilibrium point Section headings should be left justified, with ° 2a2 m2 A4 ( x , 0, z ) ® ° z x (a2  d 2 )  b2 d 2 ° d 2 m2 ¯ the first letter capitalized and numbered consecutively, starting with the Introduction. Sub-section headings should be in capital and lower-case italic letters, numbered 1.1, 1.2, etc, and left justified, with second and subsequent lines indented. You may need to insert a page break to keep a heading with its text.

3.5. A5

( x* , y * , z * )

The Jacobin matrix of A5

O 3  V 1O 2  V 2 O  V 3 V3

§ V11 V12 V13 · ¨ ¸ , The characteristic equation of ¨ V21 V22 V23 ¸ ¨V V ¸ © 31 32 V33 ¹ (V11  V22  V33 ) , V 2 (V11V33  V22V33  V11V22  V12V21  V23V32 ) ,

( x* , y* , z * ) is: J 5

0 ˈwhere V 1

(V11V22V33  V12V21V33  V11V23V32 ) . According to the Routh-Hurwitz criterion, A5

asymptotically stable if y *

x* (a1  d1 )  b1d1 ! 0, z * d1m1

J 5 is

( x* , y* , z * ) is locally and

x* (a2  d 2 )  b2 d 2 ! 0, V 1 ! 0, V 3 ! 0, V 1V 2 ! V 3 . d 2 m2

4. Bifuracation Analysis In order to obtain a more systematic investigation of the effect of mutual interference of predators and half saturation constants, we constructed bifurcation diagrams by numerical simulation. The correspondence between the eight nondimensional parameters and the original parameters was shown in Table1. Table 1 Values of the nondimensional parameters used in this study Nondimensional Parameters

Dimensional Parameters

Value used

a1

C1 A1 / R

0.3

a2

C2 A2 / R

0.2

d1

D1 / R

0.03

d2

D2 / R

0.02

b1

B1 / K

0~1

b2

B2 / K

0~1

m1

M 1 R / A1

0~1.4

m2

M 2 R / A2

0~1.4

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Jianfeng Feng et al. / Procedia Environmental Sciences 2 (2010) 42–48

To accurately calculate value of bifurcation parameters, we used software MatCont in this study. MatCont is a graphical Matlab package for the interactive numerical study of dynamical systems. The package MatCont is freely available for non-commercial use and can be downloaded from http://www.matcont.ugent.be. 4.1. Effect of b2 and m2 on the stability First, we studied the effect of b2 and m2 on the system stability. We set b2 =0.2, m1=1, and b1=0.5. We obtained bifurcation diagram of x, y variables with bifurcation parameters by using numerical simulation. Obviously, there was a Hopf bifurcation and a transcritical bifurcation in the system when m2=1.05 and 0.58, respectively (Fig. 1a, b). As b2 increased from 0.2 to 0.6, the Hopf bifurcation occurred at m2=0.35. For m2>0, system didn’t occur trans-critical bifurcation (Fig. 1c, d). However, if we increased b2 (b2=0.8) further (Fig. 1e, f), for m2 >0, system was always in a steady state. This indicates that increasing the value of b2 and m2 can enhance the system stability.

Fig. 1 Bifurcation diagrams for the model. Plots of the value of x and the value of y versus m2 for (a) the setting of b2=0.2, m1=1ˈb1=0.5, (b) the setting of b2=0.6, m1=1ˈb1=0.5, (c)the setting of b2=0.8, m1=1ˈb1=0.5. Other parameter values are as listed in Tab. 1.

4.2. Effect of b1 and m1 on the stability Then, we studied the impact of b1 and m1 on the system stability. First, we set m1=0, b2=1, m2=0 (Fig. 2a). We obtained bifurcation diagram of x, y variables and bifurcation parameters b1 by using numerical simulation. System occurred Hopf bifurcation when b1=0.83 (Fig. 2b). This means when 0< b1<0.83, point attractor in the model became a limit cycle. As m1 increased from 0 to 0.5, the Hopf bifurcation occurred at b1=0.35 and trans-critical bifurcation occurred at b1=0.51(Fig 2c, d). However, if we increased m1 (m1=1) further (Fig. 2e, f), for b1>0.02, system was always in a steady state.

Fig. 2 Bifurcation diagrams for the model. Plots of the value of x and the value of y versus b1for (a) the setting of

m1=0ˈb2=1, m2=0, (b) the

setting of m1=0.5ˈb2=1, m2=0, (c)the setting of m1=1ˈb2=1, m2=0. Other parameter values are listed in Tab. 1.

Jianfeng Feng et al. / Procedia Environmental Sciences 2 (2010) 42–48

From Fig. 2, with the increase of m1, the occurrence of the bifurcation point of bifurcation values became smaller, which means in a certain range of system, stability region is expanded. Thus, increasing the value of b1 and m1 can enhance the system stability. 5. Conclusions In this paper, we established a three-species predator-prey model with Beddington-DeAngelis functional response according to a food chain including two species of zooplanktons and one of phytoplankton, and analyzed the stability and bifurcation of this model. The influences of competitions between two zooplanktons and semisaturation constant on the stability of the system were investigated. The conclusion is that, in a certain range, increasing competition and saturation constant factor can increase the stability of the system. However, due to the complexity of the system equation, we don’t get specific bifurcation parameter value at the time of occurrence of bifurcation through mathematical analysis. So we used numerical simulation methods to investigate the specific bifurcation parameter value. On one hand, through the numerical simulation results, we could verify whether the results of using analytical method are correct. On the other hand, through numerical simulation, we find out when bifurcation happens and what type of bifurcation is. We find that the system only appears Hopf and transcritical bifurcation in a certain range. The first type of bifurcation is the Hopf bifurcation denoted H. Hopf bifurcation occurs as a pair of complex conjugate eigenvalues of the linearization around the fixed point cross the imaginary axis of the complex plane. At the Hopf bifurcation, the involved equilibrium switches from stable point to limit cycle. Although the limit cycle is also stable in the significance of ecology, Hopf bifurcations is generally interpreted as a point of destabilization [18]. The system will present periodic behavior when a stable limit cycle attractor is born. The transition from a stable equilibrium to stable periodic behavior is often connected to destabilization [18,19].

Acknowledgements Financial supports from National Natural Science Foundation of China (No.10772132), Youth Foundation of Nankai University (65010391) and National Water Project of China (2008ZX08526-003).

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