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2014, 11(2): 257-283. doi: 10.3934/mbe.2014.11.257

Stability and optimal control for some classes of tritrophic systems

Luca Galbusera 1, , Sara Pasquali 2, and  Gianni Gilioli 3,

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

Received  November 2012 Revised  August 2013 Published  October 2013

The objective of this paper is to study an optimal resource management problem for some classes of tritrophic systems composed by autotrophic resources (plants), bottom level consumers (herbivores) and top level consumers (humans). The first class of systems we discuss are linear chains, in which biomass flows from plants to herbivores, and from herbivores to humans. In the second class of systems humans are omnivorous and hence compete with herbivores for plant resources. Finally, in the third class of systems humans are omnivorous, but the plant resources are partitioned so that humans and herbivores do not complete for the same ones. The three trophic chains are expressed as Lotka-Volterra models, which seems to be a suitable choice in contexts where there is a shortage of food for the consumers. Our model parameters are taken from the literature on agro-pastoral systems in Sub-Saharan Africa.
Keywords: Lotka-Volterra, Trophic system, stability, optimal control., coexistence, extinction.
Mathematics Subject Classification: Primary: 37N35, 49-XX, 93D20; Secondary: 92Bx.
Citation: Luca Galbusera, Sara Pasquali, Gianni Gilioli. Stability and optimal control for some classes of tritrophic systems. Mathematical Biosciences & Engineering, 2014, 11 (2) : 257-283. doi: 10.3934/mbe.2014.11.257
References:
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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]

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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.

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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.

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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.

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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.

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V. S. Ivlev, Experimental Ecology of the Feeding of Fishes, Yale University Press, New Haven, Connecticut, 1961.

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V. Křivan and S. Diehl, Adaptive omnivory and species coexistence in tri-trophic food webs, Theoretical Population Biology, 67 (2005), 85-99.

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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.

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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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