I NTERNATIONAL J OURNAL OF C OMPUTATIONAL M ATHEMATICS AND NUMERICAL S IMULATION
Vol. 4, No. 1, January-June 2011, pp. 65-70,  Serials Publications, ISSN: 0973-581X
Refuge Stable Equilibrium State for Natural and
Manipulated Predators in Biological Pest Control
G. Shanmugam1 and K. B. Naidu2
ABSTRACT
In the integrated pest management (IPM) strategy, biological pest control (BPC) plays a
dominating role economically, sociologically and ecologically. Thus BPC can also be
considered as a cultural control in the process of IPM as it minimizes the environmental
hazards caused by pesticides. In this paper mathematical model for (1) control by natural
enemies (2) control by manipulated release of predators and (3) control by natural enemies
together with manipulated (released) predators are considered and compared by numerical
simulation.
1. INTRODUCTION
Conservation of natural enemies is probably the most important and readily available
biological control practice for agriculturist and gardeners. Natural enemies occur in all
areas, from the backyard garden to the commercial field. They are adapted to the local
environment and to the target pest. Their conservation is simple and cost-effective.
Lacewings, lady beetles, hover fly larvae and parasitized aphid mummies are examples of
natural enemies present in aphid colonies. Fungus-infected adult flies are also natural enemies
during periods of high humidity. Prevention of accidental eradication of these natural enemies
(using pesticides) should be avoided. This is termed as simple conservation of natural
predators.
Many commercial insectaries rear and market a variety of natural enemies including
predaceous mites, lady beetles, lacewings, preying mantids, and several species of
parasitoids. (See Fig.1)
Success with such releases requires appropriate timing (the host must be present or the
natural enemy will simply die or leave the area) and release of the correct number of natural
enemies per unit area (release rate).
1
2
Department of Mathematics, Jeppiaar Engineering College, Chennai - 600119, Tamil Nadu, India, E-mail:
[email protected]
Department of Mathematics, Sathyabama University, Chennai–600119, Tamil Nadu, India, E-mail:
[email protected]
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International Journal of Computational Mathematics and Numerical Simulation
Figure 1: Release Packs for Mass Reared Natural Enemies vary in from and Function
from Left Trichogramma Waps (M.Hoffmann), Encarsia Waps (J.Sanderson)
and Orius Bugs (j. Sanderson)
One study has estimated that a successful BPC program profits in 32:1 benefit-to-cost
ratio. The same study had shown that an average chemical pesticide program only returns
profits in the ratio of 13:1.
2. MATHEMATICAL MODEL
2.1 General Population Model
Let the population growth of a species be governed by
dN
= f (N )
dt
(1)
Where f ( N ) is a nonlinear function of N. Then equilibrium states are given by N * .
Refuge Stable Equilibrium State for Natural and Manipulated Predators in Biological Pest…
Where
f (N * ) = 0
67
(2)
Expanding f ( N ) as a Taylor series about N * , where N = N * + η we obtain
f (N ) = f (N* ) +
= 0+
( )
f ′ N* η
1!
( )
f ′ N* η
1!
+
( )
f ′′ N * η2
2!
( )
f ′′ N * η2
+
2!
+ ...
+ ...
( )
*
= f′ N η
(3)
( )
f ′( N ) > 0
*
N * is a stable equilibrium state if f ′ N < 0 ;
N * is unstable equilibrium state if
*
2.2 A Model for Natural (Slow) Predation
Ludwig et.al (1978) formulated a mathematical model for natural control of the budworm
pest by the natural predators (Birds). The population dynamics of the budworm is governed
by the equation

dN
N
= rB N  1 −
dt
 KB

(4)
 − P( N )

where the first term in the right hand side is the Malthusian term and the second term is the
logistic term. (rB is the growth rate of the budworms and KB is a carring capacity which is
related to the density of the foliage available on the trees). P(N) represents predation by
birds. Ludwig et.al used the following function for predation
BN 2
P( N ) =
( A2 + N 2 )
(5)
where A and B are constant. Then the equation (4) takes the form

N
dN
= rB N  1 −
KB
dt


BN 2
−

2
2
 A +N
(
)
(6)
Introducing the non dimensional quantities
u=
Ar
K
N
Bt
, r = B, q= B, τ=
A
B
A
A
(7)
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International Journal of Computational Mathematics and Numerical Simulation
the non dimensional form of (6) is
 u
u2
du
= ru  1 −  −
= f (u; r; q)
2
dτ
 q  1+ u
(8)
Figure 2: The Non Zero Refuge Equilibrium State is given by u = 0.8419
2.3 Model for Manipulated Bpc
Let N(t) be the size of the pest population at time t. We take for the fast predation
(scientifically manipulated and released) the function given by

 −N 2
P( N ) = P 1 − exp 
2
∈ A

 
  , 0 <∈<< 1
 
(9)
Where P, A and ∈ are positive parameters. ( ∈ depends on the intensity of the predators
released in the field).
Then the model for dynamics of pest population is given by

 −N 2
N
dN

= RN  1 −  − P 1 − exp 
2
K
dt

∈ A

 
  , 0 <∈<< 1
 
(10)
where R and K are constant, positive parameters.
We use the non-dimensionalization given by
u=
N
RA
K
tp
, r=
, q= ,τ =
A
P
A
A
(11)
Then the equation (10) becomes
 −u 2
 u  
du
= ru  1 −  − 1 − exp 
dτ
 q  
 ∈
 
  = F (u )
 
(12)
Refuge Stable Equilibrium State for Natural and Manipulated Predators in Biological Pest…
69
Solution of equation (12) is not important for our purpose. We will find out the
equilibrium states of (12) where
F (u) =
du
= 0. Then we get the equation
dτ
 −u 2
 u  
du
= ru  1 −  − 1 − exp 
dτ
 q  
 ∈
 
 = 0
 
Then u = 0 is one equilibrium states. Put u = 0 + η where | η | << 1.
Then the equation (12) becomes
As τ → ∞, η → ∞ .
dη
= rη . The solution of this equation is = ce rt , r > 0
dτ
Therefore u = 0 is unstable equilibrium state.
The other equilibrium states are obtained by solving F(u) = 0.
Using MS Excel the graph of u verses F(u) for different parameters r = 2, q = 3, ∈ = 0.1
are drawn on the computer.
Figure 3: The Non Zero Refuge Equilibrium State is given by u = 0.24368
2.4 A Model for Natural Predators Together with Manipulated Predators
The model equation for the interaction of natural and manipulated predators is given by

 −u 2
 u
u2
du
−
1
−
exp
= ru  1 −  −


2
dτ
 q  1 + u 
 ∈
 

 
Then the equilibrium states are given by

 −u 2
 u
du
u2
=
ru
1
−
−
−
1
−
exp
F(u) =




2
dτ
 q  1 + u 
 ∈
 
 = 0
 
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International Journal of Computational Mathematics and Numerical Simulation
Then u = 0 is one equilibrium states which is unstable. The other equilibrium states are
given by solving F(u) = 0 using Microsoft excel and numerical simulation.
Figure 4: The Non Zero Refuge Equilibrium State is given by u = 0.201
3. CONCLUSION
Refuge stable equilibrium state is one which keeps the pest at minimum non zero level.
From Fig.2 for natural predation, the refuge stable equilibrium state is at 0.8419.
From Fig.3 for the manipulated predation, the refuge stable equilibrium state is at
0.24368.
From Fig.4 for the interplay of natural predation and manipulated predation, the refuge
stable equilibrium state is at 0.201.
Thus we find that the refuge stable equilibrium state reduces when the predation
increases.
The agriculturist must so manipulate the predation that the pest settles at the refuge state.
REFERENCES
[1]
Murray. J. D., “Mathematical Biology”, Springer–Verlag, Berlin Heidelberg NewYork – London (1989).
[2]
Ludwig. D., Jones, D. D., Holling, C. S., “Qualitative Analysis of Insect Outbreak System the Spruce
Budworm and Forest”, J. Anim. Ecol., 47, 315-332, (1978).
[3]
Hassell, M. P., “The Dynamics of Arthropod”, Predator – Prey Systems, Princeton: Princeton University
Press, 1978.
[4]
Nisbet, R. M.: Gurney, W. S. C., Modeling Fluctuating Population, NewYork, Wiley, 1982.
[5]
U.S. Congress, Office of Technology Assessment 1995, Biologically Based Technologies for Pest Control,
OTA-ENV-636, Washinton, DC.hht://www.wws.princeton.edu/ota/ns20/year_f.html.
[6]
K. B. Naidu and G. Shanmugam, “A Mathematical Model for Scientifically Manipulated Fast Biological
Pest Control”, Proc. of Int. Conf. on Trendz in Information Science and Computing, 1, Page 433-438.