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Stability and optimal control for some classes of tritrophic systems
1. | previously at CNR, Institute of Applied Mathematics and Information Technology “Enrico Magenes”, Via E. Bassini 15, 20133 Milano, Italy |
2. | CNR, Institute of Applied Mathematics and Information Technology “Enrico Magenes”, Via E. Bassini 15, 20133 Milano, Italy |
3. | Department of Molecular and Translational Medicine, University of Brescia, Viale Europa 11, 25125 Brescia, Italy |
References:
[1] |
N. C. Apreutesei, Necessary optimality conditions for a Lotka-Volterra three species system, Math. Model. Nat. Phenom., 1 (2006), 120-135.
doi: 10.1051/mmnp:2006007. |
[2] |
N. C. Apreutesei, An optimal control problem for a prey-predator system with a general functional response, Appl. Math. Lett., 22 (2009), 1062-1065.
doi: 10.1016/j.aml.2009.01.016. |
[3] |
C. D. Becker and E. Ostrom, Human ecology and resource sustainability: The importance of institutional diversity, Annual Review of Ecology and Systematics, 26 (1995), 113-133. |
[4] |
J. C. Castilla, Coastal marine communities: Trends and perspectives from human-exclusion experiments, Trends in Ecology & Evolution, 14 (1999), 280-283.
doi: 10.1016/S0169-5347(99)01602-X. |
[5] |
K. S. Chaudhuri, A bioeconomic model of harvesting a multispecies fishery, Ecological Modelling, 32 (1986), 267-279.
doi: 10.1016/0304-3800(86)90091-8. |
[6] |
T. Christiaans, T. Eichner and R. Pethig, Optimal pest control in agriculture, J. Econom. Dynam. Control, 31 (2007), 3965-3985.
doi: 10.1016/j.jedc.2007.01.028. |
[7] |
N. J. Cossins and M. Upton, The Borana pastoral system of Southern Ethiopia, Agricultural Systems, 25 (1987), 199-218.
doi: 10.1016/0308-521X(87)90020-5. |
[8] |
T. Das, R. N. Mukherjee and K. S. Chaudhuri, Harvesting of a prey-predator fishery in the presence of toxicity, Appl. Math. Model., 33 (2009), 2282-2292.
doi: 10.1016/j.apm.2008.06.008. |
[9] |
S. Desta and D. L. Coppock, Pastoralism under pressure: Tracking system change in Southern Ethiopia, Human Ecology, 32 (2004), 465-486.
doi: 10.1023/B:HUEC.0000043516.56037.6b. |
[10] |
A. El-Gohary and M. T. Yassen, Optimal control and synchronization of Lotka-Volterra model, Chaos, Solitons and Fractals, 12 (2001), 2087-2093.
doi: 10.1016/S0960-0779(00)00023-0. |
[11] |
B. D. Fath, Distributed control in ecological networks, Ecological Modelling, 179 (2004), 235-245.
doi: 10.1016/j.ecolmodel.2004.06.007. |
[12] |
C. Feinstein and D. Luenberger, Analysis of the asymptotic behavior of optimal control trajectories: The implicit programming problem, SIAM J. Control Optim., 19 (1981), 561-585.
doi: 10.1137/0319035. |
[13] |
G. Gilioli and J. Baumgärtner, Parameter estimation for a disease transmission model on the population dynamics of Africa's Brown Ear Tick (Rhipicephalus appendiculatus, Acari: Ixodidae) and cattle infected by East Coast Fever, Bollettino di Zoologia Agraria e bachicoltura, Serie II, 41 (2009), 21-40. |
[14] |
A. P. Gutierrez and U. Regev, The bioeconomics of tri-trophic systems: Applications to invasive species, Ecological Economics, 52 (2005), 383-396. |
[15] |
C. S. Holling, The functional response of invertebrate predators to prey density, in Memoirs of the Entomological Society of Canada, Vol. 48, Ottawa, Canada, 1966.
doi: 10.4039/entm9848fv. |
[16] |
V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, Connecticut, 1961. |
[17] |
T. K. Kar and B. Ghosh, Sustainability and optimal control of an exploited prey predator system through provision of alternative food to predator, Biosystems, 109 (2012), 220-232.
doi: 10.1016/j.biosystems.2012.02.003. |
[18] |
V. Křivan and S. Diehl, Adaptive omnivory and species coexistence in tri-trophic food webs, Theoretical Population Biology, 67 (2005), 85-99. |
[19] |
V. Křivan and J. Eisner, Optimal foraging and predator-prey dynamics III, Theoretical Population Biology, 63 (2003), 269-279. |
[20] |
L. J. Lambourne, M. S. Dicko, P. Semenye and M. H. Butterworth, Animal nutrition in pastoral system research in sub-Saharan Africa, in Proceedings of the ILCA/IDRC Workshop held at ILCA, Addis Ababa, Ethiopia, 1983. |
[21] |
R. Lande, S. Engen and B.-E. Saether, Optimal harvesting, economic discounting and extinction risk in fluctuating populations, Nature, 11 (1994), 88-90. |
[22] |
A. Leung and S. Stojanovic, Optimal control for elliptic Volterra-Lotka type equations, J. Math. Anal. Appl., 173 (1993), 603-619.
doi: 10.1006/jmaa.1993.1091. |
[23] |
D. Ludwig, R. Hilborn and C. Walters, Uncertainty, resource exploitation, and conservation: Lessons from history, Science, 260 (1993), 17-36.
doi: 10.1126/science.260.5104.17. |
[24] |
L. Mariani and S. Parisi, Simulation of grazed grassland productivity in Ethiopian Highlands, in Sustainable agro-pastoral systems: Concepts, approaches and tools, CNR-IMATI, Milan, Italy, March 27, 2012. Available from: http://www.mi.imati.cnr.it/ sara/biodiversita/paginaweb.html. |
[25] |
T. Nakazawa and N. Yamamura, Community structure and stability analysis for intraguild interactions among host, parasitoid, and predator, Population Ecology, 48 (2006), 139-149.
doi: 10.1007/s10144-005-0249-5. |
[26] |
T. Namba, K. Tanabe and N. Maeda, Omnivory and stability of food webs, Ecological Complexity, 5 (2008), 73-85.
doi: 10.1016/j.ecocom.2008.02.001. |
[27] |
National Academy of Sciences, Tef, in Lost Crops of Africa: Vol. I: Grains, National Academies Press, Washington, 1996. |
[28] |
E. Neumayer, The human development index and sustainability: A constructive proposal, Ecological Economics, 39 (2001), 101-114.
doi: 10.1016/S0921-8009(01)00201-4. |
[29] |
M. M. Nyangito, N. K. R. Musimba and D. M. Nyariki, Range use and dynamics in the agropastoral system of southeastern Kenya, African Journal of Environmental Science and Technology, 2 (2008), 222-230. |
[30] |
T. Pradhan and K. S. Chaudhuri, A dynamic reaction model of a two-species fishery with taxation as a control instrument: A capital theoretic analysis, Ecological Modelling, 121 (1999), 1-16.
doi: 10.1016/S0304-3800(99)00062-9. |
[31] |
M. Rafikov, J. M. Balthazar and H. F. von Bremen, Mathematical modeling and control of population systems: Applications in biological pest control, Applied Mathematics and Computation, 200 (2008), 557-573.
doi: 10.1016/j.amc.2007.11.036. |
[32] |
U. Regev, A. P. Gutierrez, S. J. Schreiber and D. Zilbermann, Biological and economic foundations of renewable resource exploitation, Ecological Economics, 26 (1998), 227-242.
doi: 10.1016/S0921-8009(97)00103-1. |
[33] |
R. S. Reid, S. Serneels, M. Nyabenge and J. Hanson, The changing face of pastoral systems in grassland dominated ecosystem of East Africa, in Grassland of the World, 2005, 19-65. |
[34] |
Global Health Observatory Data Repository, 2012., Available from: , ().
|
[35] |
S. Sager, H. G. Bock, M. Diehl, G. Reinelt and J. P. Schlöder, Numerical methods for optimal control with binary control functions applied to a Lotka-Volterra type fishing problem, in Recent Advances in Optimization (ed. Alberto Seeger), Lecture Notes in Economics and Mathematical Systems, Vol. 563, Springer, Berlin, 2006, 269-289.
doi: 10.1007/3-540-28258-0\_17. |
[36] |
Y. Shastri and U. Diwekar, Sustainable ecosystem management using optimal control theory. I. Deterministic Systems, J. Theoret. Biol., 241 (2006), 506-521.
doi: 10.1016/j.jtbi.2005.12.014. |
[37] |
_______, Sustainable ecosystem management using optimal control theory. 2. Stochastic Systems, J. Theoret. Biol., 241 (2006), 522-532.
doi: 10.1016/j.jtbi.2005.12.013. |
[38] |
A. Sikder and A. B. Roy, Persistence of a four species food chain with full omnivory, Biosystems, 31 (1993), 39-47.
doi: 10.1016/0303-2647(93)90015-5. |
[39] |
X. Song and L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci., 170 (2001), 173-186.
doi: 10.1016/S0025-5564(00)00068-7. |
[40] |
P. D. N. Srinivasu, B. S. R. V. Prasad and M. Venkatesulu, Biological control through provision of additional food to predators: A theoretical study, Theoretical Population Biology, 72 (2007), 111-120.
doi: 10.1016/j.tpb.2007.03.011. |
[41] |
Yu. M. Svirezhev and D. O. Logofet, Stability of Biological Communities, "Mir", Moscow, 1983. |
[42] |
F. M. Wilkes, Capital Budgeting Techniques, John Wiley & Sons, New York, 1977. |
show all references
References:
[1] |
N. C. Apreutesei, Necessary optimality conditions for a Lotka-Volterra three species system, Math. Model. Nat. Phenom., 1 (2006), 120-135.
doi: 10.1051/mmnp:2006007. |
[2] |
N. C. Apreutesei, An optimal control problem for a prey-predator system with a general functional response, Appl. Math. Lett., 22 (2009), 1062-1065.
doi: 10.1016/j.aml.2009.01.016. |
[3] |
C. D. Becker and E. Ostrom, Human ecology and resource sustainability: The importance of institutional diversity, Annual Review of Ecology and Systematics, 26 (1995), 113-133. |
[4] |
J. C. Castilla, Coastal marine communities: Trends and perspectives from human-exclusion experiments, Trends in Ecology & Evolution, 14 (1999), 280-283.
doi: 10.1016/S0169-5347(99)01602-X. |
[5] |
K. S. Chaudhuri, A bioeconomic model of harvesting a multispecies fishery, Ecological Modelling, 32 (1986), 267-279.
doi: 10.1016/0304-3800(86)90091-8. |
[6] |
T. Christiaans, T. Eichner and R. Pethig, Optimal pest control in agriculture, J. Econom. Dynam. Control, 31 (2007), 3965-3985.
doi: 10.1016/j.jedc.2007.01.028. |
[7] |
N. J. Cossins and M. Upton, The Borana pastoral system of Southern Ethiopia, Agricultural Systems, 25 (1987), 199-218.
doi: 10.1016/0308-521X(87)90020-5. |
[8] |
T. Das, R. N. Mukherjee and K. S. Chaudhuri, Harvesting of a prey-predator fishery in the presence of toxicity, Appl. Math. Model., 33 (2009), 2282-2292.
doi: 10.1016/j.apm.2008.06.008. |
[9] |
S. Desta and D. L. Coppock, Pastoralism under pressure: Tracking system change in Southern Ethiopia, Human Ecology, 32 (2004), 465-486.
doi: 10.1023/B:HUEC.0000043516.56037.6b. |
[10] |
A. El-Gohary and M. T. Yassen, Optimal control and synchronization of Lotka-Volterra model, Chaos, Solitons and Fractals, 12 (2001), 2087-2093.
doi: 10.1016/S0960-0779(00)00023-0. |
[11] |
B. D. Fath, Distributed control in ecological networks, Ecological Modelling, 179 (2004), 235-245.
doi: 10.1016/j.ecolmodel.2004.06.007. |
[12] |
C. Feinstein and D. Luenberger, Analysis of the asymptotic behavior of optimal control trajectories: The implicit programming problem, SIAM J. Control Optim., 19 (1981), 561-585.
doi: 10.1137/0319035. |
[13] |
G. Gilioli and J. Baumgärtner, Parameter estimation for a disease transmission model on the population dynamics of Africa's Brown Ear Tick (Rhipicephalus appendiculatus, Acari: Ixodidae) and cattle infected by East Coast Fever, Bollettino di Zoologia Agraria e bachicoltura, Serie II, 41 (2009), 21-40. |
[14] |
A. P. Gutierrez and U. Regev, The bioeconomics of tri-trophic systems: Applications to invasive species, Ecological Economics, 52 (2005), 383-396. |
[15] |
C. S. Holling, The functional response of invertebrate predators to prey density, in Memoirs of the Entomological Society of Canada, Vol. 48, Ottawa, Canada, 1966.
doi: 10.4039/entm9848fv. |
[16] |
V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, Connecticut, 1961. |
[17] |
T. K. Kar and B. Ghosh, Sustainability and optimal control of an exploited prey predator system through provision of alternative food to predator, Biosystems, 109 (2012), 220-232.
doi: 10.1016/j.biosystems.2012.02.003. |
[18] |
V. Křivan and S. Diehl, Adaptive omnivory and species coexistence in tri-trophic food webs, Theoretical Population Biology, 67 (2005), 85-99. |
[19] |
V. Křivan and J. Eisner, Optimal foraging and predator-prey dynamics III, Theoretical Population Biology, 63 (2003), 269-279. |
[20] |
L. J. Lambourne, M. S. Dicko, P. Semenye and M. H. Butterworth, Animal nutrition in pastoral system research in sub-Saharan Africa, in Proceedings of the ILCA/IDRC Workshop held at ILCA, Addis Ababa, Ethiopia, 1983. |
[21] |
R. Lande, S. Engen and B.-E. Saether, Optimal harvesting, economic discounting and extinction risk in fluctuating populations, Nature, 11 (1994), 88-90. |
[22] |
A. Leung and S. Stojanovic, Optimal control for elliptic Volterra-Lotka type equations, J. Math. Anal. Appl., 173 (1993), 603-619.
doi: 10.1006/jmaa.1993.1091. |
[23] |
D. Ludwig, R. Hilborn and C. Walters, Uncertainty, resource exploitation, and conservation: Lessons from history, Science, 260 (1993), 17-36.
doi: 10.1126/science.260.5104.17. |
[24] |
L. Mariani and S. Parisi, Simulation of grazed grassland productivity in Ethiopian Highlands, in Sustainable agro-pastoral systems: Concepts, approaches and tools, CNR-IMATI, Milan, Italy, March 27, 2012. Available from: http://www.mi.imati.cnr.it/ sara/biodiversita/paginaweb.html. |
[25] |
T. Nakazawa and N. Yamamura, Community structure and stability analysis for intraguild interactions among host, parasitoid, and predator, Population Ecology, 48 (2006), 139-149.
doi: 10.1007/s10144-005-0249-5. |
[26] |
T. Namba, K. Tanabe and N. Maeda, Omnivory and stability of food webs, Ecological Complexity, 5 (2008), 73-85.
doi: 10.1016/j.ecocom.2008.02.001. |
[27] |
National Academy of Sciences, Tef, in Lost Crops of Africa: Vol. I: Grains, National Academies Press, Washington, 1996. |
[28] |
E. Neumayer, The human development index and sustainability: A constructive proposal, Ecological Economics, 39 (2001), 101-114.
doi: 10.1016/S0921-8009(01)00201-4. |
[29] |
M. M. Nyangito, N. K. R. Musimba and D. M. Nyariki, Range use and dynamics in the agropastoral system of southeastern Kenya, African Journal of Environmental Science and Technology, 2 (2008), 222-230. |
[30] |
T. Pradhan and K. S. Chaudhuri, A dynamic reaction model of a two-species fishery with taxation as a control instrument: A capital theoretic analysis, Ecological Modelling, 121 (1999), 1-16.
doi: 10.1016/S0304-3800(99)00062-9. |
[31] |
M. Rafikov, J. M. Balthazar and H. F. von Bremen, Mathematical modeling and control of population systems: Applications in biological pest control, Applied Mathematics and Computation, 200 (2008), 557-573.
doi: 10.1016/j.amc.2007.11.036. |
[32] |
U. Regev, A. P. Gutierrez, S. J. Schreiber and D. Zilbermann, Biological and economic foundations of renewable resource exploitation, Ecological Economics, 26 (1998), 227-242.
doi: 10.1016/S0921-8009(97)00103-1. |
[33] |
R. S. Reid, S. Serneels, M. Nyabenge and J. Hanson, The changing face of pastoral systems in grassland dominated ecosystem of East Africa, in Grassland of the World, 2005, 19-65. |
[34] |
Global Health Observatory Data Repository, 2012., Available from: , ().
|
[35] |
S. Sager, H. G. Bock, M. Diehl, G. Reinelt and J. P. Schlöder, Numerical methods for optimal control with binary control functions applied to a Lotka-Volterra type fishing problem, in Recent Advances in Optimization (ed. Alberto Seeger), Lecture Notes in Economics and Mathematical Systems, Vol. 563, Springer, Berlin, 2006, 269-289.
doi: 10.1007/3-540-28258-0\_17. |
[36] |
Y. Shastri and U. Diwekar, Sustainable ecosystem management using optimal control theory. I. Deterministic Systems, J. Theoret. Biol., 241 (2006), 506-521.
doi: 10.1016/j.jtbi.2005.12.014. |
[37] |
_______, Sustainable ecosystem management using optimal control theory. 2. Stochastic Systems, J. Theoret. Biol., 241 (2006), 522-532.
doi: 10.1016/j.jtbi.2005.12.013. |
[38] |
A. Sikder and A. B. Roy, Persistence of a four species food chain with full omnivory, Biosystems, 31 (1993), 39-47.
doi: 10.1016/0303-2647(93)90015-5. |
[39] |
X. Song and L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure, Math. Biosci., 170 (2001), 173-186.
doi: 10.1016/S0025-5564(00)00068-7. |
[40] |
P. D. N. Srinivasu, B. S. R. V. Prasad and M. Venkatesulu, Biological control through provision of additional food to predators: A theoretical study, Theoretical Population Biology, 72 (2007), 111-120.
doi: 10.1016/j.tpb.2007.03.011. |
[41] |
Yu. M. Svirezhev and D. O. Logofet, Stability of Biological Communities, "Mir", Moscow, 1983. |
[42] |
F. M. Wilkes, Capital Budgeting Techniques, John Wiley & Sons, New York, 1977. |
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