American Institute of Mathematical Sciences

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

Mathematical modeling of Glassy-winged sharpshooter population

Jeong-Mi Yoon 1, , Volodymyr Hrynkiv 1, , Lisa Morano 2, , Anh Tuan Nguyen 3, , Sara Wilder 3, and  Forrest Mitchell 4,

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.
Keywords: Pierce's disease, Mathematical modeling, Glassy-winged sharpshooter..
Mathematics Subject Classification: Primary: 92D25, 92D40; Secondary: 93C2.
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. Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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. Google Scholar

[8]

, ., (). Google Scholar

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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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[15]

, ., (). Google Scholar

[16]

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

[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. Google Scholar

[18]

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

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[29]

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

[30]

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

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. Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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. Google Scholar

[8]

, ., (). Google Scholar

[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. Google Scholar

[10]

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

[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. Google Scholar

[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. Google Scholar

[13]

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

[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. Google Scholar

[15]

, ., (). Google Scholar

[16]

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

[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. Google Scholar

[18]

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

[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. Google Scholar

[20]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[26]

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

[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. Google Scholar

[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. Google Scholar

[29]

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

[30]

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

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