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Keywords:

  • Sterile pest;
  • fertile pest;
  • sterile insect technique;
  • sterile insect release rate (SIRR);
  • stability;
  • comparison argument;
  • global stability

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The model system
  5. 3. Equilibrium points and local stability study
  6. 4. Global behavior
  7. 5. Numerical results
  8. 6. Concluding remarks
  9. Acknowledgments
  10. REFERENCES

The menace of insect pests is a topic of major concern throughout the world. Chemical pesticides are conventionally used to control these insect pests. However, the adverse effects of these synthetic pesticides, such as high toxicity from residues in food, contamination of water and the environment resulting in human health hazard and resistance of the pest to the pesticides have necessitated development of some nonconventional approaches of biological pest control.

In this research, we have focused on a mathematical model of biological pest control using the sterile insect release technique. Unlike most of the existing modeling studies in this field that mainly deal with the pest population only, we have incorporated the crop population as a distinct dynamical equation together with the fertile and sterile insect pests. Local stability analysis is performed around the crop and fertile insect free axial equilibrium, the fertile-insect-free boundary equilibrium, the crop-free boundary equilibrium and the equilibrium point of coexistence. From the study we have derived a number of thresholds for the SIRR (the main parameter for our study) that cause existence and or extinction of the crop population as well as the fertile insect pests. A global study of the model system using comparison arguments revealed existence of a global attractor for the system. Numerical simulations are done to support and augment analytical results.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The model system
  5. 3. Equilibrium points and local stability study
  6. 4. Global behavior
  7. 5. Numerical results
  8. 6. Concluding remarks
  9. Acknowledgments
  10. REFERENCES

Rapid progress in international trade and expansion of travel and tourism together with the phenomenon of global climate change had caused an explosion of insect pests and disease vectors that threaten bio-security in almost all countries. Chemical pesticides are normally used to control these insect pests. However, the adverse effects of these chemical pesticides, namely, high toxicity, residues in food, contamination of water and the environment resulting in human health hazard are becoming topics of growing public concern [Kabir 2001]. These chemical pesticides also cause outbreak of secondary pests resulting in additional loss of crops. Moreover, these insecticides harm beneficiary insects like honey bees and other pollinators and negatively affects fish, birds and mammals.

In view of the above, there was a growing need to develop some alternative pest management strategies that are more environment-friendly [Tan and Chen 2009]. One such technique is the sterile insect technique, commonly known as SIT. In this technology, ionizing radiation is used to effectively sterilize insects without affecting the ability of the males to function in the field and mate with wild female insects. These sterile insects are then released into the environment in an area-specific way [Knipling 1979, 1984]. Mating of a wild female with a sterile male will not produce any off-spring as the generated eggs will not fertilize. It is therefore a type of birth control procedure in which the wild female insects of the pest population are not capable to reproduce when they are inseminated by released radiation-sterilized males. Consequently, most of the matings will be sterile as a large number of sterile insects are introduced into the environment. This in turn will reduce the number of native insects and at the same time increase the number of sterile insects. This ultimately drives the wild native population to extinction. SIT was pioneered in the 1950s by Dr. R.C. Bushland and Dr. E.F. Knipling who jointly received the 1992 World Food Prize [Bushland and Knipling 1992].

The first SIT program was started in 1954 in the island of Curacao to control the New World screw worm, Cochilomyia hominivorax, a parasite fly that lays its eggs in the living tissue of warm-blooded animals such as live-stocks and even humans [Hendrichs 2000; Hendrichs et al. 2002]. In South Africa, SIT was implemented to successfully control Bactrocera tryoni, commonly known as The Queensland fruit fly [Meats 1996]. The technique was implemented in Japan to remove the sweet potato pests biologically known as Cylas Formicarius and Euscepes Postfsciatus [Moriya and Miyatake 2001]. Suckling et al. [2002] reported eradication of Teia Anortoides (popularly known as the Painted Apple Moth) in New Zealand by applying this technique. With these successful implementations, the utility of this technique in removing harmful pests is gaining momentum throughout the world day by day [Meats 1996; Suckling 2003; Dyck et al. 2005; Meats et al. 2006; Vreysen et al. 2006; Hendrichs et al. 2007; White et al. 2010].

2. The model system

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The model system
  5. 3. Equilibrium points and local stability study
  6. 4. Global behavior
  7. 5. Numerical results
  8. 6. Concluding remarks
  9. Acknowledgments
  10. REFERENCES

We formulate a pest control model implementing the SIT using the following three differential equations

  • display math(1)

with initial conditions

  • display math(2)

The model consists of three population variables—inline image is the population density of sterile insects maintained within a population concentration inline image of fertile insects at a certain time t. In the absence of the sterile insects, the birth term of the (fertile) insect population will be inline image with r2 as the birth rate. Introduction of sterile insects will alter the birth term to inline image. It is also assumed that the sterile and the fertile insects have the same density independent death rate [Murray 1993] which is taken as d. The density dependent death rates due to self-interaction between the fertile and or sterile insects is taken as α and due to interaction between the sterile and fertile insects is taken as β [Lewis and Van den Driessche 1993]. δ is the constant release rate for sterile insects. The third population variable inline image represents the crop population on which these insect pests feed. As, the dynamical evolution of the insect pests are closely linked with that of their prey, namely, the crop population, we have included the crop dynamics in our modeling study. r1 is taken as the growth rate and k the carrying capacity of the crop. a1 and a2 represent the invasion rate of the fertile and the sterile insects with e1 and e2 as the corresponding conversion efficiencies. The invasion is assumed to follow a Holling-type II functional response.

The following lemma ensures the positivity and boundedness of the solutions of system (1).

Lemma 1. For system (1), all the solutions starting from the first quadrant will remain positive and bounded.

Proof. Let, inline image. Then from (1)

  • display math

which implies

  • display math

where inline image. Applying the theory of differential inequalities, one can write

  • display math(3)

Hence for inline image,

  • display math(4)

which shows that all the solutions initiated in inline image will remain in the region

  • display math

3. Equilibrium points and local stability study

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The model system
  5. 3. Equilibrium points and local stability study
  6. 4. Global behavior
  7. 5. Numerical results
  8. 6. Concluding remarks
  9. Acknowledgments
  10. REFERENCES

The system possesses the following equilibrium points:

  1. The axial equilibrium inline image.
  2. The crop-free boundary equilibrium inline image where inline image and S1 is the positive root of the equation
    • display math(5)

    Note 1. The axial equilibrium E0 will always exist while the boundary equilibrium E1 will exist for large values of δ, the sterile insect release rate (SIRR).

  3. The fertile insect-free boundary equilibrium inline image where inline image and S2 is the positive root of the equation
    • display math(6)
  4. The equilibrium point of coexistence inline image where inline image, inline image and inline image are the positive solutions of the system of equations
  • display math(7)

A local stability study around E0 reveals that E0 will be locally asymptotically stable (LAS) if inline image which implies that

  • display math(8)

Therefore, the crop population will become extinct if the release rate of the sterile insect crosses a certain threshold, namely inline image.

We now perform a local stability analysis of (1) around the crop-free boundary point E1. The characteristic equation of the Jacobian matrix around E1 is

  • display math(9)

where

  • display math(10)

Now, equation (9) will have roots with positive real parts if inline image and inline image, which is possible when α is very small and

  • display math(11)

Thus the crop-free boundary steady state will become unstable when r1 satisfies (11) and α is very small.

Next we study the fertile pest-free equilibrium E2. The characteristic equation corresponding to the Jacobian matrix around E2 is given by

  • display math(12)

where

  • display math(13)

Now, E3 will be unstable if inline image which reduces to

  • display math(14)

Thus it is seen that when the parameter β (which arises from the density dependent death rate of the pest) is below a certain threshold, the fertile pest-free equilibrium point will exhibit unstable behavior.

Finally, we consider the interior equilibrium point of species coexistence. The characteristic equation of the Jacobian matrix around inline image is given by

  • display math(15)

where

  • display math(16)

Now, inline image will be locally asymptotically stable if inline image, inline image and inline image.

Due to the complexity in the algebraic expressions involved, we postpone a detailed analytical study of this equilibrium point. Instead we will perform a numerical study around this steady state later.

4. Global behavior

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The model system
  5. 3. Equilibrium points and local stability study
  6. 4. Global behavior
  7. 5. Numerical results
  8. 6. Concluding remarks
  9. Acknowledgments
  10. REFERENCES

In this section, we will study the global behavior of the model system by using the comparison theorem [Xu and Ma 2009].

Theorem 1. Let inline image be a positive solution of the system of equations

  • display math

Then inline image will be a global attractor for the system (1).

Proof. Assuming inline image, we have already shown in (4) that inline image. Hence it follows that

  • display math(17)

Consequently, for small inline image, there exists inline image such that inline image for all inline image. So from the second equation of (1) we can write for inline image

  • display math(18)

Then the auxiliary equation of (18) will be given by

  • display math

which gives

  • display math(19)

Now, applying comparison theorem [Xu and Ma 2009] and letting inline image, we have

  • display math(20)

Again from the third equation of (1), for inline image,

  • display math(21)

By using the same method, we obtain from (21)

  • display math(22)

From (22), it is seen that for small ε, there exists inline image such that for all inline image,

  • display math

and using comparison theorem [Xu and Ma 2009]

  • display math

Again using comparison theorem [Xu and Ma 2009], we get,

  • display math

Similarly,

  • display math

Therefore for small inline image, there exists a inline image such that for all inline image inline image, inline image and inline image.

Therefore,

  • display math

So,

  • display math

Therefore,

  • display math

Applying comparison theorem [Xu and Ma 2009] and letting inline image, we get

  • display math(23)

Since inline image and inline image, inline image provided inline image. Consequently, we can form a sequence inline image which is monotonically decreasing and bounded below. Following the same argument, we define the following sequences which are also monotonically decreasing and bounded below since each element of inline image, inline image, and inline image is positive.

  • display math(24)

Consequently, all of these sequences are convergent.

Let, inline image; inline image; and inline image.

Again, inline image, inline image and inline image are monotonically increasing as inline image, inline image, and inline image are monotonically decreasing and bounded above since inline image, inline image, and inline image for all n.

Let, inline image, inline image, and inline image.

Then, we have the following set of equations:

  • display math(25)

Now, inline image will converge to zero for small d as inline image is increasing. So inline image. Let inline image be a positive solution of the system of equations

  • display math(26)

Then inline image and inline image will converge to inline image; inline image and inline image will converge to inline image whereas inline image will converge to inline image. Consequently, inline image will be a global attractor for the system (1). This completes the proof.inline image

5. Numerical results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The model system
  5. 3. Equilibrium points and local stability study
  6. 4. Global behavior
  7. 5. Numerical results
  8. 6. Concluding remarks
  9. Acknowledgments
  10. REFERENCES

In this section, we perform a numerical simulation study of the model equations around the various equilibrium points for a range of parameter values. The parameter values are taken mainly from the existing literature [Lewis and Van den Driessche 1993; Dyck et al. 2005; Maiti et al. 2006; Tan and Chen 2009]. The birth rate r2 of the fertile insect pests usually resides in the range inline image offsprings per individual per day. In particular, the tse-tse flies lies at the lower end of the scale while the fruit flies are at the higher end. The density independent death rate d lies in the range inline image [Lewis and Van den Driessche 1993]. Similarly, the SIRR δ also varies over a large possible range of values. We have numerically studied the dynamics of the system for a wide variation in these parameter values. In the following table, we show the parameter values that we finally select through our numerical study and which best explains the results that we have obtained from our mathematical analysis. Using these parameter values, we first evaluate the crop and fertile insect-free axial equilibrium inline image. The threshold SIRR inline image as deduced in (8) is also evaluated as inline image. In Figure 1, we draw the population time graph for inline image with inline image and find that after a certain time, the crop population density will increase from the zero equilibrium level and become stable at a certain nonzero density level. While for inline image with inline image, the crop population vanishes after a certain time [Figure 2].

Table 1. Parameter values
ParameterDescriptionUnitsValues
r1Intrinsic growth rate of cropper hectare per day5
kEnvironmental carrying capacity of cropNumber per unit area20
α, βDensity dependent death rates for insectshectare per insect per day0.3
r2Intrinsic birth rate of fertile insectper hectare per day20
dDensity independent death rate of insectper hectare per day0.6
a1Crop consumption rate of fertile pestper hectare per day0.3
a2Crop consumption rate of sterile pestper hectare per day0.3
mHalf saturation constantNumber per unit area0.8
inline imageConversion efficiency of insectper hectare per day0.5
δRelease rate of sterile insectsper hectare per dayVariable
image

Figure 1. Numerical simulation of the model around the axial state inline image for inline image. The parameter values are given in the table with inline image.

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image

Figure 2. Numerical simulation of the model around the axial state inline image for inline image. The parameter values are given in the table with inline image.

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Secondly, we obtain the crop-free boundary point E1 as inline image. Simulation study around this equilibrium reveals that for a certain release rate of the sterile population (inline image), the crop population will exist while the fertile pest will vanish. But if the release rate δ is increased (inline image), the crop population will become extinct [Figures 3 and 4]. Here we have assumed small values for α (inline image) and r1 satisfying (11).

image

Figure 3. Numerical simulation of the model around the boundary state inline image. The parameter values are given in the table with inline image and inline image showing existence of the crop population.

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image

Figure 4. Numerical simulation of the model around the boundary state inline image. The parameter values are given in the table with inline image and inline image showing extinction of the crop population.

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Next we consider the boundary point E2 which is fertile insect-free. We evaluate inline image. We have performed simulations around this point for various values of δ and found that a variation in the SIRR has no effect on the time evolution of the crop population [Figure 5]. Finally, we have experimented with the equilibrium point of coexistence inline image. Our experiment showed the existence of a critical SIRR inline image such that for inline image (inline image), all component populations will co-exist while for inline image, the fertile pest will undergo extinction and the crop will coexist with the sterile pest. This is shown in Figures 6 and 7. In Figure 8, we draw the phase plot of the model system around the interior state and obtained the existence of an attractor.

image

Figure 5. Numerical simulation of the model around the boundary state inline image. The parameter values are given in the table with inline image and inline image.

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image

Figure 6. Numerical simulation of the model around the interior state inline image for inline image showing stable coexistence. The parameter values are given in the table with inline image.

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image

Figure 7. Numerical simulation of the model around the interior state inline image for inline image showing fertile insect extinction. The parameter values are given in the table with inline image.

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image

Figure 8. Phase plot of the model around the interior state. The figure exhibits the existence of global attractor.

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6. Concluding remarks

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The model system
  5. 3. Equilibrium points and local stability study
  6. 4. Global behavior
  7. 5. Numerical results
  8. 6. Concluding remarks
  9. Acknowledgments
  10. REFERENCES

The warfare between man and the insect pests is now more than thousand years old. A significant portion of the postharvest loss of agricultural products all over the world is due to destruction of crops by insect pests. Chemical pesticides are increasingly used to destroy these insect pests. In fact, world pesticide consumption is increasing at the rate of 5% per year [Hendrichs 2000]. However, this overuse of pesticides negatively affects the environment and subsequently the resident fish, bird and mammals. Moreover, indiscriminate use of these pesticides have caused outbreak of secondary pests, which results in additional loss of crop population. These side effects have necessitated alternative methods of pest control that is effective in controlling pests and at the same time reduce the risk of side effects.

In this research, we have studied a mathematical model of pest control that incorporates such an alternative pest management strategy commonly known as sterile insect (release) technique. Mathematical modeling study utilizing this technique has been previously performed by many researchers in different contexts [Lewis 1993; Meats 1996, 2006; Meats et al. 2006; Maiti et al. 2006]. However, most of these studies have dealt with the fertile and sterile insect pests only and the crop population on which these pests have a remarkable influence has been ignored. One of the most important aspects of insect pest control is to maintain the density of pests below the economic threshold level. This level, in turn, is closely connected with the dynamical evolution of the crop population. Accordingly, while formulating the model equations, we have included the dynamics of the crop population together with that of the sterile and the fertile insect pests. We have identified a number of ecologically important equilibria for the system: (i) the crop and fertile insect-free axial state E0; (ii) the crop-free boundary state E1; (iii) the fertile insect-free boundary state E2; and (iv) the coexistent interior state inline image. As is already stated, our main aim is to investigate the threshold parameter values that ensure survival of the crop population. From our analytical study of the axial equilibrium point E0, we have deduced a critical value inline image of the SIRR in equation (8), such that as δ crosses inline image, the crop population will become extinct. We have also validated this finding through our numerical simulations in Figures 1 and 2. Analysis of the boundary point E1 revealed that when the density dependent death rate α is small, the existence and or extinction criteria of the crop will again depend on the SIRR in a way that an increase in δ will cause crop extinction. However, for the other boundary state E2, where the fertile pest is not present, the SIRR has no effect on crop survival. For the equilibrium point of co-existence inline image, we have not performed a detailed analytical study due to complexity of algebraic expressions. Instead, we have carried out a numerical experimentation on this equilibrium point and obtained another threshold value of sterile release rate namely, δ1 (mentioned in Section 'Numerical results'), such that when inline image, all the component populations will coexist around inline image and as δ crosses δ1, the fertile pest will undergo extinction while the crop will survive with the sterile pest species. Finally, we have investigated the global behavior of the model equations using comparison theorem [Xu and Ma 2009; Bhattacharyya and Mukhopadhyay 2011]. The investigation demonstrated the existence of a global attractor for the system where all the populations coexist and towards which the system will converge after a long time.

Summarizing the above analysis, we conclude that the SIRR δ is a very crucial parameter that have a remarkable influence in determining the existence and or extinction of the crop population as well as in controlling the population density of fertile pests. The various critical thresholds for this release rate that we have obtained from our analytical and numerical study could be utilized by resource managers in managing insect pests and devising various important pest management strategies.

REFERENCES

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. The model system
  5. 3. Equilibrium points and local stability study
  6. 4. Global behavior
  7. 5. Numerical results
  8. 6. Concluding remarks
  9. Acknowledgments
  10. REFERENCES
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