• Previous Article
    Synaptic energy drives the information processing mechanisms in spiking neural networks
  • MBE Home
  • This Issue
  • Next Article
    On the return process with refractoriness for a non-homogeneous Ornstein-Uhlenbeck neuronal model
2014, 11(2): 257-283. doi: 10.3934/mbe.2014.11.257

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

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.
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:
[1]

N. C. Apreutesei, Necessary optimality conditions for a Lotka-Volterra three species system,, Math. Model. Nat. Phenom., 1 (2006), 120. doi: 10.1051/mmnp:2006007. Google Scholar

[2]

N. C. Apreutesei, An optimal control problem for a prey-predator system with a general functional response,, Appl. Math. Lett., 22 (2009), 1062. doi: 10.1016/j.aml.2009.01.016. Google Scholar

[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. Google Scholar

[4]

J. C. Castilla, Coastal marine communities: Trends and perspectives from human-exclusion experiments,, Trends in Ecology & Evolution, 14 (1999), 280. doi: 10.1016/S0169-5347(99)01602-X. Google Scholar

[5]

K. S. Chaudhuri, A bioeconomic model of harvesting a multispecies fishery,, Ecological Modelling, 32 (1986), 267. doi: 10.1016/0304-3800(86)90091-8. Google Scholar

[6]

T. Christiaans, T. Eichner and R. Pethig, Optimal pest control in agriculture,, J. Econom. Dynam. Control, 31 (2007), 3965. doi: 10.1016/j.jedc.2007.01.028. Google Scholar

[7]

N. J. Cossins and M. Upton, The Borana pastoral system of Southern Ethiopia,, Agricultural Systems, 25 (1987), 199. doi: 10.1016/0308-521X(87)90020-5. Google Scholar

[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. doi: 10.1016/j.apm.2008.06.008. Google Scholar

[9]

S. Desta and D. L. Coppock, Pastoralism under pressure: Tracking system change in Southern Ethiopia,, Human Ecology, 32 (2004), 465. doi: 10.1023/B:HUEC.0000043516.56037.6b. Google Scholar

[10]

A. El-Gohary and M. T. Yassen, Optimal control and synchronization of Lotka-Volterra model,, Chaos, 12 (2001), 2087. doi: 10.1016/S0960-0779(00)00023-0. Google Scholar

[11]

B. D. Fath, Distributed control in ecological networks,, Ecological Modelling, 179 (2004), 235. doi: 10.1016/j.ecolmodel.2004.06.007. Google Scholar

[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. doi: 10.1137/0319035. Google Scholar

[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, 41 (2009), 21. Google Scholar

[14]

A. P. Gutierrez and U. Regev, The bioeconomics of tri-trophic systems: Applications to invasive species,, Ecological Economics, 52 (2005), 383. Google Scholar

[15]

C. S. Holling, The functional response of invertebrate predators to prey density,, in Memoirs of the Entomological Society of Canada, (1966). doi: 10.4039/entm9848fv. Google Scholar

[16]

V. S. Ivlev, Experimental Ecology of the Feeding of Fishes,, Yale University Press, (1961). Google Scholar

[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. doi: 10.1016/j.biosystems.2012.02.003. Google Scholar

[18]

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

[19]

V. Křivan and J. Eisner, Optimal foraging and predator-prey dynamics III,, Theoretical Population Biology, 63 (2003), 269. Google Scholar

[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, (1983). Google Scholar

[21]

R. Lande, S. Engen and B.-E. Saether, Optimal harvesting, economic discounting and extinction risk in fluctuating populations,, Nature, 11 (1994), 88. Google Scholar

[22]

A. Leung and S. Stojanovic, Optimal control for elliptic Volterra-Lotka type equations,, J. Math. Anal. Appl., 173 (1993), 603. doi: 10.1006/jmaa.1993.1091. Google Scholar

[23]

D. Ludwig, R. Hilborn and C. Walters, Uncertainty, resource exploitation, and conservation: Lessons from history,, Science, 260 (1993), 17. doi: 10.1126/science.260.5104.17. Google Scholar

[24]

L. Mariani and S. Parisi, Simulation of grazed grassland productivity in Ethiopian Highlands,, in Sustainable agro-pastoral systems: Concepts, (2012). Google Scholar

[25]

T. Nakazawa and N. Yamamura, Community structure and stability analysis for intraguild interactions among host, parasitoid, and predator,, Population Ecology, 48 (2006), 139. doi: 10.1007/s10144-005-0249-5. Google Scholar

[26]

T. Namba, K. Tanabe and N. Maeda, Omnivory and stability of food webs,, Ecological Complexity, 5 (2008), 73. doi: 10.1016/j.ecocom.2008.02.001. Google Scholar

[27]

National Academy of Sciences, Tef,, in Lost Crops of Africa: Vol. I: Grains, (1996). Google Scholar

[28]

E. Neumayer, The human development index and sustainability: A constructive proposal,, Ecological Economics, 39 (2001), 101. doi: 10.1016/S0921-8009(01)00201-4. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0304-3800(99)00062-9. Google Scholar

[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. doi: 10.1016/j.amc.2007.11.036. Google Scholar

[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. doi: 10.1016/S0921-8009(97)00103-1. Google Scholar

[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. Google Scholar

[34]

Global Health Observatory Data Repository, 2012., Available from: , (). Google Scholar

[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), (2006), 269. doi: 10.1007/3-540-28258-0\_17. Google Scholar

[36]

Y. Shastri and U. Diwekar, Sustainable ecosystem management using optimal control theory. I. Deterministic Systems,, J. Theoret. Biol., 241 (2006), 506. doi: 10.1016/j.jtbi.2005.12.014. Google Scholar

[37]

_______, Sustainable ecosystem management using optimal control theory. 2. Stochastic Systems,, J. Theoret. Biol., 241 (2006), 522. doi: 10.1016/j.jtbi.2005.12.013. Google Scholar

[38]

A. Sikder and A. B. Roy, Persistence of a four species food chain with full omnivory,, Biosystems, 31 (1993), 39. doi: 10.1016/0303-2647(93)90015-5. Google Scholar

[39]

X. Song and L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure,, Math. Biosci., 170 (2001), 173. doi: 10.1016/S0025-5564(00)00068-7. Google Scholar

[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. doi: 10.1016/j.tpb.2007.03.011. Google Scholar

[41]

Yu. M. Svirezhev and D. O. Logofet, Stability of Biological Communities,, 1983., (1983). Google Scholar

[42]

F. M. Wilkes, Capital Budgeting Techniques,, John Wiley & Sons, (1977). Google Scholar

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. doi: 10.1051/mmnp:2006007. Google Scholar

[2]

N. C. Apreutesei, An optimal control problem for a prey-predator system with a general functional response,, Appl. Math. Lett., 22 (2009), 1062. doi: 10.1016/j.aml.2009.01.016. Google Scholar

[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. Google Scholar

[4]

J. C. Castilla, Coastal marine communities: Trends and perspectives from human-exclusion experiments,, Trends in Ecology & Evolution, 14 (1999), 280. doi: 10.1016/S0169-5347(99)01602-X. Google Scholar

[5]

K. S. Chaudhuri, A bioeconomic model of harvesting a multispecies fishery,, Ecological Modelling, 32 (1986), 267. doi: 10.1016/0304-3800(86)90091-8. Google Scholar

[6]

T. Christiaans, T. Eichner and R. Pethig, Optimal pest control in agriculture,, J. Econom. Dynam. Control, 31 (2007), 3965. doi: 10.1016/j.jedc.2007.01.028. Google Scholar

[7]

N. J. Cossins and M. Upton, The Borana pastoral system of Southern Ethiopia,, Agricultural Systems, 25 (1987), 199. doi: 10.1016/0308-521X(87)90020-5. Google Scholar

[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. doi: 10.1016/j.apm.2008.06.008. Google Scholar

[9]

S. Desta and D. L. Coppock, Pastoralism under pressure: Tracking system change in Southern Ethiopia,, Human Ecology, 32 (2004), 465. doi: 10.1023/B:HUEC.0000043516.56037.6b. Google Scholar

[10]

A. El-Gohary and M. T. Yassen, Optimal control and synchronization of Lotka-Volterra model,, Chaos, 12 (2001), 2087. doi: 10.1016/S0960-0779(00)00023-0. Google Scholar

[11]

B. D. Fath, Distributed control in ecological networks,, Ecological Modelling, 179 (2004), 235. doi: 10.1016/j.ecolmodel.2004.06.007. Google Scholar

[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. doi: 10.1137/0319035. Google Scholar

[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, 41 (2009), 21. Google Scholar

[14]

A. P. Gutierrez and U. Regev, The bioeconomics of tri-trophic systems: Applications to invasive species,, Ecological Economics, 52 (2005), 383. Google Scholar

[15]

C. S. Holling, The functional response of invertebrate predators to prey density,, in Memoirs of the Entomological Society of Canada, (1966). doi: 10.4039/entm9848fv. Google Scholar

[16]

V. S. Ivlev, Experimental Ecology of the Feeding of Fishes,, Yale University Press, (1961). Google Scholar

[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. doi: 10.1016/j.biosystems.2012.02.003. Google Scholar

[18]

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

[19]

V. Křivan and J. Eisner, Optimal foraging and predator-prey dynamics III,, Theoretical Population Biology, 63 (2003), 269. Google Scholar

[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, (1983). Google Scholar

[21]

R. Lande, S. Engen and B.-E. Saether, Optimal harvesting, economic discounting and extinction risk in fluctuating populations,, Nature, 11 (1994), 88. Google Scholar

[22]

A. Leung and S. Stojanovic, Optimal control for elliptic Volterra-Lotka type equations,, J. Math. Anal. Appl., 173 (1993), 603. doi: 10.1006/jmaa.1993.1091. Google Scholar

[23]

D. Ludwig, R. Hilborn and C. Walters, Uncertainty, resource exploitation, and conservation: Lessons from history,, Science, 260 (1993), 17. doi: 10.1126/science.260.5104.17. Google Scholar

[24]

L. Mariani and S. Parisi, Simulation of grazed grassland productivity in Ethiopian Highlands,, in Sustainable agro-pastoral systems: Concepts, (2012). Google Scholar

[25]

T. Nakazawa and N. Yamamura, Community structure and stability analysis for intraguild interactions among host, parasitoid, and predator,, Population Ecology, 48 (2006), 139. doi: 10.1007/s10144-005-0249-5. Google Scholar

[26]

T. Namba, K. Tanabe and N. Maeda, Omnivory and stability of food webs,, Ecological Complexity, 5 (2008), 73. doi: 10.1016/j.ecocom.2008.02.001. Google Scholar

[27]

National Academy of Sciences, Tef,, in Lost Crops of Africa: Vol. I: Grains, (1996). Google Scholar

[28]

E. Neumayer, The human development index and sustainability: A constructive proposal,, Ecological Economics, 39 (2001), 101. doi: 10.1016/S0921-8009(01)00201-4. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0304-3800(99)00062-9. Google Scholar

[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. doi: 10.1016/j.amc.2007.11.036. Google Scholar

[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. doi: 10.1016/S0921-8009(97)00103-1. Google Scholar

[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. Google Scholar

[34]

Global Health Observatory Data Repository, 2012., Available from: , (). Google Scholar

[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), (2006), 269. doi: 10.1007/3-540-28258-0\_17. Google Scholar

[36]

Y. Shastri and U. Diwekar, Sustainable ecosystem management using optimal control theory. I. Deterministic Systems,, J. Theoret. Biol., 241 (2006), 506. doi: 10.1016/j.jtbi.2005.12.014. Google Scholar

[37]

_______, Sustainable ecosystem management using optimal control theory. 2. Stochastic Systems,, J. Theoret. Biol., 241 (2006), 522. doi: 10.1016/j.jtbi.2005.12.013. Google Scholar

[38]

A. Sikder and A. B. Roy, Persistence of a four species food chain with full omnivory,, Biosystems, 31 (1993), 39. doi: 10.1016/0303-2647(93)90015-5. Google Scholar

[39]

X. Song and L. Chen, Optimal harvesting and stability for a two-species competitive system with stage structure,, Math. Biosci., 170 (2001), 173. doi: 10.1016/S0025-5564(00)00068-7. Google Scholar

[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. doi: 10.1016/j.tpb.2007.03.011. Google Scholar

[41]

Yu. M. Svirezhev and D. O. Logofet, Stability of Biological Communities,, 1983., (1983). Google Scholar

[42]

F. M. Wilkes, Capital Budgeting Techniques,, John Wiley & Sons, (1977). Google Scholar

[1]

Georg Hetzer, Tung Nguyen, Wenxian Shen. Coexistence and extinction in the Volterra-Lotka competition model with nonlocal dispersal. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1699-1722. doi: 10.3934/cpaa.2012.11.1699

[2]

S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system. Mathematical Biosciences & Engineering, 2006, 3 (1) : 173-187. doi: 10.3934/mbe.2006.3.173

[3]

Yasuhisa Saito. A global stability result for an N-species Lotka-Volterra food chain system with distributed time delays. Conference Publications, 2003, 2003 (Special) : 771-777. doi: 10.3934/proc.2003.2003.771

[4]

Zhaohai Ma, Rong Yuan, Yang Wang, Xin Wu. Multidimensional stability of planar traveling waves for the delayed nonlocal dispersal competitive Lotka-Volterra system. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2069-2092. doi: 10.3934/cpaa.2019093

[5]

Tung Nguyen, Nar Rawal. Coexistence and extinction in Time-Periodic Volterra-Lotka type systems with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3799-3816. doi: 10.3934/dcdsb.2018080

[6]

Dan Li, Jing'an Cui, Yan Zhang. Permanence and extinction of non-autonomous Lotka-Volterra facultative systems with jump-diffusion. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2069-2088. doi: 10.3934/dcdsb.2015.20.2069

[7]

Francisco Montes de Oca, Liliana Pérez. Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2663-2690. doi: 10.3934/dcdsb.2015.20.2663

[8]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[9]

Yuan Lou, Dongmei Xiao, Peng Zhou. Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 953-969. doi: 10.3934/dcds.2016.36.953

[10]

Linping Peng, Zhaosheng Feng, Changjian Liu. Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4807-4826. doi: 10.3934/dcds.2014.34.4807

[11]

Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559

[12]

Fuke Wu, Yangzi Hu. Stochastic Lotka-Volterra system with unbounded distributed delay. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 275-288. doi: 10.3934/dcdsb.2010.14.275

[13]

Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083

[14]

Qi Wang, Chunyi Gai, Jingda Yan. Qualitative analysis of a Lotka-Volterra competition system with advection. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1239-1284. doi: 10.3934/dcds.2015.35.1239

[15]

Anthony W. Leung, Xiaojie Hou, Wei Feng. Traveling wave solutions for Lotka-Volterra system re-visited. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 171-196. doi: 10.3934/dcdsb.2011.15.171

[16]

Qi Wang, Yang Song, Lingjie Shao. Boundedness and persistence of populations in advective Lotka-Volterra competition system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2245-2263. doi: 10.3934/dcdsb.2018195

[17]

Juan Luis García Guirao, Marek Lampart. Transitivity of a Lotka-Volterra map. Discrete & Continuous Dynamical Systems - B, 2008, 9 (1) : 75-82. doi: 10.3934/dcdsb.2008.9.75

[18]

Xiaoling Zou, Ke Wang. Optimal harvesting for a stochastic N-dimensional competitive Lotka-Volterra model with jumps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 683-701. doi: 10.3934/dcdsb.2015.20.683

[19]

Xiao He, Sining Zheng. Protection zone in a modified Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2027-2038. doi: 10.3934/dcdsb.2015.20.2027

[20]

Yoshiaki Muroya. A Lotka-Volterra system with patch structure (related to a multi-group SI epidemic model). Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 999-1008. doi: 10.3934/dcdss.2015.8.999

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (18)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]