2014, 11(3): 667-677. doi: 10.3934/mbe.2014.11.667

Mathematical modeling of Glassy-winged sharpshooter population

1. 

Department of Mathematics and Statistics, University of Houston - Downtown, Houston, TX 77002, United States, United States

2. 

Department of Natural Sciences, University of Houston - Downtown, Houston, TX 77002, United States

3. 

University of Houston - Downtown, Houston, TX 77002, United States, United States

4. 

Department of Entomology, Texas A&M AgriLife Research, Stephenville, TX 76401, United States

Received  January 2013 Revised  July 2013 Published  January 2014

Pierce's disease (PD) is a fatal disease of grapevines which results from an infection by the plant pathogen Xyllela fastidiosa. This bacterium grows in the xylem (water-conducting) vessels of the plant blocking movement of water. PD can kill vines in one year and poses a serious threat to both the California and the expanding Texas wine industries. Bacteria are vectored from one vine to the next by a number of xylem feeding insect species. Of these, the Glassy-winged Sharpshooter (GWSS) is considered to be the primary xylem feeding insect in Texas vineyards. An extensive database of the xylem-feeding population frequencies was collected by USDA-APHIS for Texas vineyards over multiple years. This project focused on a subset of data, GWSS frequencies within 25 vineyards in Edwards Plateau located in central Texas. The proposed model investigates the natural population dynamics and the decline in GWSS, likely the result of pest management campaigns on the insects within the region. The model is a delay Gompertz differential equation with harvesting and immigration terms, and we use the data to estimate the model parameters.
Citation: Jeong-Mi Yoon, Volodymyr Hrynkiv, Lisa Morano, Anh Tuan Nguyen, Sara Wilder, Forrest Mitchell. Mathematical modeling of Glassy-winged sharpshooter population. Mathematical Biosciences & Engineering, 2014, 11 (3) : 667-677. doi: 10.3934/mbe.2014.11.667
References:
[1]

R. Almeida, et al., Vector trasmission of Xylella fastidiosa: Applying fundamental knowledge to generate disease management strategies,, Ann. Entomol. Soc. Am., 98 (2005), 775.

[2]

M. Begon, J. Harper and C. Townsend, Ecology: Individuals, Populations and Communities,, 2nd edition, (1990).

[3]

M. Blua, P. Philips,and R. A. Redak, A new sharpshooter threatens both crops and ornamentals,, Calif. Agr., 53 (1999), 22. doi: 10.1094/PHP-2000-0627-01-RS.

[4]

S. Choi and N. Koo, Oscillation theory for delay and neutral differential equations,, Trends Math., 2 (1999), 170.

[5]

D. R. Causton and J. C. Venus, The Biometry of Plant Growth,, Edward Arnold, (1981). doi: 10.1111/j.1365-3040.1978.tb00759.x.

[6]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics,, Lecture Notes in Biomathematics 20, (1977).

[7]

J. De Leon, W. Joses and D. Morgan, Population genetic structure of Homalodisca coagulata (Homoptera: Cicadellidae), the vector of the bacterium Xylella fastidiosa causing Pierce's disease in grapevines,, Ann. Entomol. Soc. Am., 97 (2004), 809.

[8]

, ., ().

[9]

W. W. Fox, An exponential surplus yield model for optimizing in exploited fish populations,, T. Am. Fish. Soc., 99 (1970), 80. doi: 10.1577/1548-8659(1970)99<80:AESMFO>2.0.CO;2.

[10]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics,, Mathematics and its Applications 74, (1992).

[11]

J. Grandgirard, G. Roderick, N. Davies, M. S. Hoddle and J. N. Petit, Engineering an invasion: Classical biological control of the glassy-winged sharpshooter, Homalodisca vitripennis, by the egg parasitoid Gonatocerus ashmeadi in Tahiti and Moorea, French Polynesia,, Biol. Invasions, 10 (2008), 135. doi: 10.1007/s10530-007-9116-y.

[12]

N. Hayes, Roots of the transcendendal equation associated with a certain differential difference equation,, J. London Math. Soc., 25 (1950), 226. doi: 10.1112/jlms/s1-25.3.226.

[13]

W. Hewitt, The probable home of Pierce's disease virus,, Plant Dis. Rep., 42 (1958), 211.

[14]

C. B. Hutchinson, Circular causal systems in ecology,, Ann. N.Y. Acad. Sci., 50 (1948), 221. doi: 10.1111/j.1749-6632.1948.tb39854.x.

[15]

, ., ().

[16]

M. Kot, Elements of Mathematical Ecology,, Cambridge University Press, (2001). doi: 10.1017/CBO9780511608520.

[17]

R. Krugner, J. R. Hagler, J. G. Morse, A. P. Flores, R. L. Groves and M. W. Johnson, Seasonal population dynamics of Homalodisca vitripennis (Hemiptera: Cicadellidae) in sweet orange trees maintained under continuous deficit irrigation,, J. Econ. Entomol., 102 (2009), 960. doi: 10.1603/029.102.0315.

[18]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993).

[19]

I. M. Lauziére, S. Sheather and F. L. Mitchell, Seasonal abundance and spatio-temporal distribution of dominant xylem fluid-feeding Hemiptera in vineyards of central Texas and surrounding habitats,, Environ. Entomol., 37 (2008), 925.

[20]

N. MacDonald, Time Lags in Biological Models,, Lecture Notes in Biomathematics 27, (1978).

[21]

F. L. Mitchell, J. Brady, B. Bextine and I. M. Lauziére, Seasonal increase of Xylella fastidiosa in Hemiptera collected from central Texas vineyards,, J. Econ. Entomol., 102 (2009), 1743. doi: 10.1603/029.102.0503.

[22]

L. Morano, J. Yoon, A. Abedi and F. Mitchell, Evaluation of xylem-feeding insects (Hemiptera: Auchennorrhyncha) in Texas vineyards: distribution along state-wide environmental gradients,, Southwest. Entomol., 35 (2010), 503. doi: 10.3958/059.035.0402.

[23]

R. Redak, et al., The biology of xylem fluid-feeding insect vectors of Xylella fastidiosa and their relation to disease epidemiology,, Annu. Rev. Entomol., 49 (2004), 243. doi: 10.1146/annurev.ento.49.061802.123403.

[24]

S. Ruan, Delay differential equations in single species dynamics,, in Delay Differential Equations and Applications (eds. O. Arino et al.), (2006), 477. doi: 10.1007/1-4020-3647-7_11.

[25]

M. Setamou and W. A. Jones, Biology and biometry of sharpshooter Homalodisca coagulata (Homoptera: Cicadellidae) reared on cowpea,, Ann. Entomol. Soc. Am., 98 (2005), 322.

[26]

J. Maynard Smith, Models in Ecology,, Cambridge University Press, (1974).

[27]

Hal Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences,, Springer, (2011). doi: 10.1007/978-1-4419-7646-8.

[28]

D. Takiya, S. McKamey and R. Cavichioli, Validity of Homalodisca and of H. vitripennis as the name for glassy-winged sharpshooter (Hemiptera: Cicadellidae),, Ann. Entomol. Soc. Am., 99 (2006), 648.

[29]

T. E. Wheldon, Mathematical Models in Cancer Research,, Adam Hilger, (1988).

[30]

E. M. Wright, The non-linear difference-differential equation,, Q. J. Math., 17 (1946), 245.

show all references

References:
[1]

R. Almeida, et al., Vector trasmission of Xylella fastidiosa: Applying fundamental knowledge to generate disease management strategies,, Ann. Entomol. Soc. Am., 98 (2005), 775.

[2]

M. Begon, J. Harper and C. Townsend, Ecology: Individuals, Populations and Communities,, 2nd edition, (1990).

[3]

M. Blua, P. Philips,and R. A. Redak, A new sharpshooter threatens both crops and ornamentals,, Calif. Agr., 53 (1999), 22. doi: 10.1094/PHP-2000-0627-01-RS.

[4]

S. Choi and N. Koo, Oscillation theory for delay and neutral differential equations,, Trends Math., 2 (1999), 170.

[5]

D. R. Causton and J. C. Venus, The Biometry of Plant Growth,, Edward Arnold, (1981). doi: 10.1111/j.1365-3040.1978.tb00759.x.

[6]

J. M. Cushing, Integrodifferential Equations and Delay Models in Population Dynamics,, Lecture Notes in Biomathematics 20, (1977).

[7]

J. De Leon, W. Joses and D. Morgan, Population genetic structure of Homalodisca coagulata (Homoptera: Cicadellidae), the vector of the bacterium Xylella fastidiosa causing Pierce's disease in grapevines,, Ann. Entomol. Soc. Am., 97 (2004), 809.

[8]

, ., ().

[9]

W. W. Fox, An exponential surplus yield model for optimizing in exploited fish populations,, T. Am. Fish. Soc., 99 (1970), 80. doi: 10.1577/1548-8659(1970)99<80:AESMFO>2.0.CO;2.

[10]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics,, Mathematics and its Applications 74, (1992).

[11]

J. Grandgirard, G. Roderick, N. Davies, M. S. Hoddle and J. N. Petit, Engineering an invasion: Classical biological control of the glassy-winged sharpshooter, Homalodisca vitripennis, by the egg parasitoid Gonatocerus ashmeadi in Tahiti and Moorea, French Polynesia,, Biol. Invasions, 10 (2008), 135. doi: 10.1007/s10530-007-9116-y.

[12]

N. Hayes, Roots of the transcendendal equation associated with a certain differential difference equation,, J. London Math. Soc., 25 (1950), 226. doi: 10.1112/jlms/s1-25.3.226.

[13]

W. Hewitt, The probable home of Pierce's disease virus,, Plant Dis. Rep., 42 (1958), 211.

[14]

C. B. Hutchinson, Circular causal systems in ecology,, Ann. N.Y. Acad. Sci., 50 (1948), 221. doi: 10.1111/j.1749-6632.1948.tb39854.x.

[15]

, ., ().

[16]

M. Kot, Elements of Mathematical Ecology,, Cambridge University Press, (2001). doi: 10.1017/CBO9780511608520.

[17]

R. Krugner, J. R. Hagler, J. G. Morse, A. P. Flores, R. L. Groves and M. W. Johnson, Seasonal population dynamics of Homalodisca vitripennis (Hemiptera: Cicadellidae) in sweet orange trees maintained under continuous deficit irrigation,, J. Econ. Entomol., 102 (2009), 960. doi: 10.1603/029.102.0315.

[18]

Y. Kuang, Delay Differential Equations with Applications in Population Dynamics,, Academic Press, (1993).

[19]

I. M. Lauziére, S. Sheather and F. L. Mitchell, Seasonal abundance and spatio-temporal distribution of dominant xylem fluid-feeding Hemiptera in vineyards of central Texas and surrounding habitats,, Environ. Entomol., 37 (2008), 925.

[20]

N. MacDonald, Time Lags in Biological Models,, Lecture Notes in Biomathematics 27, (1978).

[21]

F. L. Mitchell, J. Brady, B. Bextine and I. M. Lauziére, Seasonal increase of Xylella fastidiosa in Hemiptera collected from central Texas vineyards,, J. Econ. Entomol., 102 (2009), 1743. doi: 10.1603/029.102.0503.

[22]

L. Morano, J. Yoon, A. Abedi and F. Mitchell, Evaluation of xylem-feeding insects (Hemiptera: Auchennorrhyncha) in Texas vineyards: distribution along state-wide environmental gradients,, Southwest. Entomol., 35 (2010), 503. doi: 10.3958/059.035.0402.

[23]

R. Redak, et al., The biology of xylem fluid-feeding insect vectors of Xylella fastidiosa and their relation to disease epidemiology,, Annu. Rev. Entomol., 49 (2004), 243. doi: 10.1146/annurev.ento.49.061802.123403.

[24]

S. Ruan, Delay differential equations in single species dynamics,, in Delay Differential Equations and Applications (eds. O. Arino et al.), (2006), 477. doi: 10.1007/1-4020-3647-7_11.

[25]

M. Setamou and W. A. Jones, Biology and biometry of sharpshooter Homalodisca coagulata (Homoptera: Cicadellidae) reared on cowpea,, Ann. Entomol. Soc. Am., 98 (2005), 322.

[26]

J. Maynard Smith, Models in Ecology,, Cambridge University Press, (1974).

[27]

Hal Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences,, Springer, (2011). doi: 10.1007/978-1-4419-7646-8.

[28]

D. Takiya, S. McKamey and R. Cavichioli, Validity of Homalodisca and of H. vitripennis as the name for glassy-winged sharpshooter (Hemiptera: Cicadellidae),, Ann. Entomol. Soc. Am., 99 (2006), 648.

[29]

T. E. Wheldon, Mathematical Models in Cancer Research,, Adam Hilger, (1988).

[30]

E. M. Wright, The non-linear difference-differential equation,, Q. J. Math., 17 (1946), 245.

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