Lotka-Volterra systems with retarded argument

Book summary

Summary of the book…

Book cover

Ch.1. Preliminaries
1.1. Process of mathematical modeling
1.1.1. Lotka-Volterra model
1.1.2. Lotka-Volterra model with delay
1.2. Systems of differential equations with  retarded argument
1.3. Theorems  of fixed point on metric spaces
1.4. Weakly Picard operators on metric spaces
1.5. Spline functions

Ch.2. Existence and uniqueness theorems and date dependence
2.1. Formulation of the problem
2.2. Step method
2.3. Existence and uniqueness
2.4. Comparison theorems
2.5. Continuous dependence with respect to the dates of the problems
2.6. Data dependence of the fixed point set
2.7. Examples
2.8. Remarks and generalizations

Ch.3. Differentiability with respect to parameters
3.1. Formulation of the problem
3.2. Continuous differentiability with respect to parameters
3.3. Continuous differentiability with respect to delays
3.4. Differentiability with respect to parameters via step method
3.5. Examples

Ch.4 Methods of approximation for the solution
4.1. The quadratures formula techniques
4.1.1. The quadratures formula techniques
4.1.2. Formulation of the problem
4.1.3. The numerical of the integral from the succesive approximation
4.1.4. The approximation of the succesive approximation sequences
4.2. Spline functions techniques
4.2.1. Numerical solutions with spline functions of even degree
4.2.3. Examples






see the expanding block below

Cite this book as:

Author, title, publisher, year.

Book Title

Lotka-Volterra systems with retarded argument


Cluj University Press

Print ISBN



Diana Otrocol
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Online ISBN

Google scholar

The book on google scholar.

[1] O. Agratini, Gh. Coman, I. Chiorean, R. Trîmbitaş, Analiză numerică şi teoria aproximării, Vol.3, Presa Universitară Clujeană, Cluj-Napoca, 2002.

[2] S. Ahmad, A. C. Lazer, Average growth and extinction in a competitive Lotka-Volterra system, Nonlinear Analysis, 62 (2005), no. 3, 545-557.

[3] S. Ahmad, A. C. Lazer, Average growth and total permanence in a competitive Lotka-Volterra system, Ann. Mat. Pura Appl. (4), 185 (2006), 47-67.

[4] H. Akça, Gh. Micula, Numerical solutions of system of differential equation with deviating argument by spline functions, Itinerant seminar of functional equations approximation and convexity, Cluj Napoca, 1990.

[5] H. Amann, Ordinary differential equations, Berlin, New York, 1990.

[6] N. C. Apreutesei, Asymptotic approximations and optimal control for Lotka-Volterra systems, Sci. Ann. Univ. Agric. Sci. Vet. Med., 47 (2004), no. 2, 79–85.

[7] J. Arino, P. van den Driessche, Time delays in epidemic models, modeling and numerical aspects, preprint, Proceedings of Marrakech 2002 Conference ‘School on Delay Differential Equations and Applications’, A CISD-CIMPA-UNESCO-MOROCCO School and A NATO Advanced Study Institute, September 9-21 2002, Marrakech (Morroco).

[8] R. A. Armstrong, R. McGehee, Competitive exclusion, Amer. Naturalist, 115 (1980), 151-170.

[9] C.T.H. Baker, C.A.H. Paul, Pitfalls in parameter estimation for delay differential   equations, Numerical Analysis Report, no. 267, 1995.

[10] V. Barbu, Ecuaţii diferenţiale, Ed Junimea, Iaşi, 1985.

[11]  H. Bereketoglu, I. Györi, Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay, Journal of Mathematical Analysis and Applications, 210 (1997), 279-291.

[12] E. Beretta, Y. Kuang, Convergence results in a well-known delayed predator-prey system, Journal of Mathematical Analysis and Applications, 204 (1996), 840-853.

[13] V. Berinde, Contracţii generalizate şi aplicaţii, teză de doctorat, Univ. “Babes-Bolyai” Cluj-Napoca, 1993.

[14]  P. Blaga, Some even degree spline interpolation, Studia Univ. “Babeş-Bolyai”, Mathematica, 37 (1992), no. 1, 65-72.

[15]  P. Blaga, Gh. Micula, Polynomial natural spline functions of even degree, Studia Univ.”Babeş-Bolyai”, Mathematica 38 (1993), no. 2, 31-40.

[16]  P. Blaga, Gh. Micula, Polynomial spline functions of even degree approximating the solutions of differential equations, Analele Universităţii din Timişoara, 36 (1989), 2, 171-190.

[17]  P. Blaga, Gh. Micula, H. Akça, On the use of spline functions of even degree for the numerical solution of the delay differential equations, Calcolo 32 (), no. 1-2, 83-101.

[18]  N. Boccara, Modeling Complex Systems, New-York, Springer, 2004.

[19]  C. de Boor, A practical guide to splines, Springer Verlag, New York, 1978.

[20]  F. Brauer, Some applications of the theory of ordinary differential equations to population growth problems, An. Acad. Brasil. Ciènc., 48 (1976), no. 3, 369-385.

[21]  F. Brauer, C. Castillo-Chávez, Mathematical models in population biology and epidemiology, New-York, Springer, 2001.

[22]  G. J. Butler, H. I. Freedman, P. Waltman, Uniformly persistent systems, Proc. Am. Math. Soc., 96 (1986), 425-430.

[23]  G. J. Butler, P. Waltman, Persistence in dynamical systems, J. Diff. Equations, 86 (1986), 225-263.

[24]  F. Calio, E. Marchetti, R. Pavani, About the deficient spline collocation method for particular differential and integral equations with delay, Rend. Sem. Mat. Univ. Pol. Torino, Vol. 61 (2003), no. 3, 287-300.

[25]  V. A. Căuş, Gh. Micula, Numerical solutions of the delay differential equations by nonpolynomial spline functions, Studia Univ. “Babeş-Bolyai”, Informatica, 46 (2001), no. 2, 91-98.

[26]  D. M. Chan, J. E. Franke, Probabilities of extinction, weak extinction, permanence, and mutual exclusion in discrete , competitive, Lotka-Volterra systems that involve invading species, Math. Comput. Modelling, 40 (2004), no. 7-8, 809-821.

[27]  F. Chen, Positive periodic solutions of neutral Lotka-Volterra system with feedback control, Appl. Math. Comput., 162 (2005), no. 3, 1279-1302.

[28]  F. Chen, Global asymptotic stability in n species nonautonomous Lotka-Volterra competitive systems with infinite delays and feedback control, Appl. Math. Comput., 170 (2005), no. 2, 1452-1468.

[29] F. Chen, The permanence and global attractivity of Lotka-Volterra competition system with feedback controls, Nonlinear Anal. Real World Appl., 7 (2006), no. 1, 133-143.

[30]  C. Coles, D. Murio, Numerical solutions of inverse spatial Lotka-Volterra systems, Math. Comput. Modelling, 42 (2005), no. 13, 1411-1420.

[31]  Gh. Coman, Analiză Numerică, Ed. Libris, Cluj-Napoca,1995.

[32]  Gh. Coman, G. Pavel, I. Rus, I. A. Rus, Introducere în teoria ecuaţiilor operatoriale, Editura Dacia, Cluj-Napoca, 1976.

[33]  J. Cui, M. Guo, Permanence in logistic and Lotka-Volterra systems with dispersal and time delays, Electron. J. Differential Equations (2005), no. 60, 11pp.

[34]  V. A. Darzu, Data dependence for functional differential equations of mixed types, Mathematica, 46 (69), (2004), no. 1, 61-22.

[35]  V. Dincuţă, An application of the weakly Picard operators technique to a Dirichlet problem, Seminar on Fixed Point Theory, 1 (2000), 35-38.

[36]  C. Egami, N. Hirano, Periodic solutions in a class of periodic delay predator-prey systems, Yokahama Mathematical Journal, Vol. 51 (2004).

[37]  M. M. A. El-Sheikh, S. A. A. El-Mahrouf, On stability and bifurcation of periodic solutions of delay differential equations, J. Appl. Math.& Computing, 19 (2005), no. 1-2, 281-295.

[38]  W. H. Enright, H. Hayashi, A delay differential equation solver based on a continuous Runge Kutta method with defect control, Numerical Algorithms, 16 (1997), 349-364.

[39]  T. Faria, Stability for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.

[40]  J.P. Finerty, The population ecology of cycles in small mammals, New Haven: Yale University Press, 1980.

[41]  H. I. Freedman, S. Ruan, Uniform persistence in functional differential equations, J. Differential Equations, 115 (1995).

[42]  H. I. Freedman, P. Waltman, Persistence in models of three interacting predator-prey population, Math. Biosc., 68 (1984), 213-231.

[43]  M. E. Gilpin, Do Hares Eat Lynx?, American Naturalist 107 (1973), 727-730.

[44]  C. I. Gheorghiu, Metode numerice pentru sisteme dinamice, Scientific Book Publishing House, Cluj-Napoca, 2004.

[45]  S. A. Gourley, Y. Kuang, A stage structure predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188-200.

[46]  I. Györi, Some mathematical aspects of modelling cell population dynamics, Computers Math. Applic. 20 (1997), no. 4-6, 127-138.

[47]  I. Györi, On approximation of the solutions of delay differential equations by using piecewise constant arguments, Internat. J. Math. & Math. Sci., 14 (1991), no. 1, 111-126.

[48]  E. Hairer, S. P. Norsett, G. Wanner, Solving ordinary differential equations, I, Springer Verlag, 1992.

[49]  A. Halanay, Teoria calitativă a ecuaţiilor diferenţiale, Ed. Academiei, Bucureşti, 1963.

[50]  J. Hale, Functional differential equations, Springer Verlag, New-York, Heidelberg, Berlin, 1977.

[51]  J. Hale, L. Ladeira, Differentiability with respect to delays, J. Differential Equations, 92 (1991), 14–26.

[52]  P. Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New-York, London, Sydney, 1964.

[53]  X.-Z. He, Stability and delays in a predator-prey system, J. Math. Anal. Appl. 198 (1996), 355-370.

[54]  J. Hofbauer, General cooperation theorem for hypercycles, Monatsheft Math., 91 (1981), 223-240.

[55]  J. Hofbauer, J. W-H. So, Diagonal dominance and harmless off-diagonal delays, Proc. Amer. Math. Soc. 128 (2000), 2675-2682.

[56]  V. M. Hokkanen, Gh. Moroşanu, Differentiability with respect to delay, Differential Integral Equations, 11 (1998), no. 4, 589-603.

[57]  C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the european pine sawfly, Canadian Entomologist 91 (1959) 293-320.

[58]  C. S. Holling, The functional response of predators to prey density and its role in mimicry and population, Memoirs of the Entomological Society of Canada 45 (1965) 1-60.

[59]  C. Iancu, A numerical method for a approximating the solution of an integral equation from biomathematics, Studia Univ. “Babes-Bolyai”, Mathematica, 43 (1998), no. 4 37-45.

[60]  C. Iancu, Modelare matematică. Teme speciale, Casa Cărţii de Ştiinţă, Cluj Napoca, 2002.

[61]  R. Illner, C. S. Bohun, S. McCollum, T. van Roode, Mathematical Modelling. A Case Study Approach, Student Mathematical Library, Vol. 27, American Mathematical Society, 2005.

[62]  D. V. Ionescu, Cuadraturi numerice, Ed. Tehnică, Bucureşti, 1957.

[63]  L. Jiang, Z. Zhan, Existence of positive periodic solutions for cooperative Lotka-Volterra difference system with variable delays, Math. Sci. Res. J., 9 (2005), no. 9, 244-254.

[64]  Z. Jin, H. Maoan, L. Guihua, The persistence in a Lotka-Volterra competition systems with impulsive, Chaos Solitons Fractals, 24 (2005), no. 4, 1105-1117.

[65]  H. Jing, Z. Yang, The impact of state feedback control on a predator-prey model with functional response, Discrete and Continuous Dynamical Systems, Series B, Vol 4, no. 3 (2004), 607-614.

[66]  Y. Kan-On, Bifurcation structure of positive stationary solutions for a Lotka-Volterra competition model with diffusion. II. Global structure. Dicrete Contin. Dyn. Syst., 14 (2006), no. 1, 135-148.

[67]  T. K. Kar, U. K. Pahari, K. S. Chaudhuri, Conservation of a prey-predator fishery with predator self limitation based on continuous fishing effort, J. Appl. Math. & Computing, 19 (2005), nos. 1-2, 311-326.

[68]  Y. Kuang, Basic properties of mathematical population models, J. of Biomathematics, 17 (2002) 129-142.

[69]  Y. Kuang, Global stability in delay differential system without dominating instantaneous negative feedbacks, J. Differential Equations 119 (1995), no. 2, 503-532.

[70]  J. D. Lambert, Computational methods in ordinary differential equations, John Wiley & Sons, 1972.

[71]  D. Lacidignola, C. Tebaldi, Effects of ecological differentiation on Lotka-Volterra systems for species with behavioral adaptation and variable growth rates, Math. Biosci, 194 (2005), no. 1, 95-123.

[72]  Y. Li, Y. Kuang, Periodic solutions in periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 255 (2001a), 260-280.

[73]  Y. Li, Y. Kuang, Periodic solutions in periodic delayed Gause-type predator-prey systems, Proceeding of Dynamic Systems and Applications, 3 (2001b), 375-382.

[74]  Y. Li, Y. Kuang, Periodic solutions in periodic state-dependent delay equations and population models, Proceedings of the American Mathematical Society, 130 (2001), no. 5, 1345-1352.

[75]  M. Li, Y. Duan, W. Zhang, M. Wang, The existence of positive periodic solutions of a class of Lotka-Volterra type impulsive systems with infinitely distributed delay, Comput. Math. Appl., 49 (2005), no. 7-8, 1037-1044.

[76]  J. Li, J. Shen, Positive periodic solutions of impulsive Lotka-Volterra equations, Ann. Differential Equations, 21 (2005), no. 2, 161-168.

[77]  Y. Li, L. Zhu, Existence of periodic solutions of discrete Lotka-Volterra systems with delays, Bull. Inst. Math. Acad. Sinica, 33 (2005), no. 4, 369-380.

[78]  S. Lin, Z. Lu, Permanence for two-species Lotka-Volterra systems with delays, Math. Biosci. Eng, 3 (2006), no. 1, 137-144.

[79]  W. Lin, T. Chen, Positive periodic solutions of delayed periodic Lotka-Volterra systems, Phys. Lett. A. 334 (2005), no. 4, 273-287.

[80]  B. Liu, Y. Zhang, L. Chen, The dynamical behaviors of a Lotka-Volterra predator-prey model concerning integrated pest management, Nonlinear Anal. Real World Appl., 6 (2005), no. 2, 227–243.

[81]  S. Liu, L. Chen, Permanence, extinction and balancing survival in nonautonomous Lotka-Volterra system with delays, Appl. Math. Comp., 129 (2002), 481-499.

[82]  F. R. Loscalzo, T.D. Talbot, Spline function approximations for solutions of ordinary differential equations, Bull. Amer. Math. Soc., 73 (1967), 438-442.

[83]  A. J. Lotka, Elements of Physical Biology, Baltimore, Williams and Wilkins, 1925. Reprinted under the new title: Elements of Mathematical Biology, New York, Dover, 1956.

[84]  Z. Y. Lu, Y. Takeuchi, Permanence and global attractivity for competitive Lotka-Volterra systems with delay, Nonlinear Analysis, 22 (1994), no. 7, 847-856.

[85]  Z. Lu, W. Wang, Global stability for two species Lotka-Volterra systems with delays, J. Math. Anal. Appl. 208 (1997), 277-280.

[86]  W. Ma, Y. Takeuchi, Stability analysis on a predator-prey system with distributed delays, J. Comp. Appl. Math., 88 (1998), 79–94.

[87]  R. M. May, Stability and complexity in model ecosystems, Princeton University Press, Princeton, N.J., 1974.

[88]  X. Meng, Almost periodic solution for a class of Lotka-Volterra type N species ecological systems with time delay, J. Syst. Sci. Complex, 18 (2005), no. 4, 488-497.

[89]  X. Meng, T. Zhang, H. Liu, Asymptotic property for a Lotka-Volterra competitive system with delays and dispersion, Ann. Differential Equations, 21 (2005), no. 3, 378-384.

[90]  Gh. Micula, Funcţii spline şi aplicaţii, Editura Tehnică, Bucureşti, 1978.

[91]  Gh. Micula, S. Micula, Handbook of Splines, Kluwer Academic Publishers, Netherland, 1999.

[92]  H. N. Moreira, W. Yuquan, A Lyapunov function for an alternative Lotka-Volterra competition model, Trabalhos de Matematica, nr. 278, 1993.

[93]  V. Mureşan, Ecuaţii diferenţiale cu modificarea afină a argumentului, Transilvania Press, Cluj Napoca, 1997.

[94]  V. Mureşan, Functional integral equation, Media Mira, Cluj -Napoca, 2003.

[95]  V. Mureşan, On some functional-integral equations with modification of the argument, Seminar on Fixed Point Theory Cluj-Napoca, 3 (2002), 297-304.

[96]  Y. Muroya, Uniform persistence for Lotka-Volterra-type delay differential systems, Nonlinear Analysis, 4 (2003).

[97]  Y. Muroya, Contractivity and global stability for discrete models of Lotka-Volterra type, Hokkaido Math. J., 34 (2005), no. 2, 277–297.

[98]  Y. Muroya, Partial survival and extinction of species in discrete nonautonomous Lotka-Volterra systems, Tokyo L. Math, 28 (2005), no. 1, 189-200.

[99]  Y. Muroya, Boundeness and partial survival of species in nonautonomous Lotka-Volterra systems, Nonlinear Anal. Real World Appl., 6 (2005), no. 2, 263-272.

[100]  J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, Heidelberg, New-York, 1989.

[101]  S. Nakaoka, Y. Saito, Y. Takeuchi, Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system, Mathematical Biosciences. Engineering, 3 (2006), no. 1, 173-187.

[102]  D. Otrocol, A numerical method for approximating the solution of a Lotka-Volterra system with two delays, Studia Univ. “Babeş-Bolyai”, Mathematica, 50 (2005), no. 1, 99-110.

[103]  D. Otrocol, Data dependence for the solution of a Lotka-Volterra system with two delays, Mathematica, 48 (71), (2006), no. 1, 61-68.

[104]  D. Otrocol, Lotka-Volterra system with two delays, via weakly Picard operators, Nonlinear Analysis Forum, 10(2) (2005), 193-199.

[105]  D. Otrocol, Numerical solutions of Lotka-Volterra system with delay by spline functions of even degree, Studia Univ. “Babeş-Bolyai”, Mathematica, 51 (2006), no. 4, 167-180.

[106]  D. Otrocol, Smooth dependence on parameters for a differential equation with delay from population dynamics, 2007 International Conference on Engineering and Mathematics, Bilbao, Spain, July 9-11, 2007, 3-10.

[107]  D. Otrocol, Differentiability with respect to a parameter for a Lotka-Volterra system with delays, via step method, Revue d’Analyse Numérique de l’Approximation, 35 (2006), no. 1, 83-86.

[108]  D. Otrocol, Data dependence of the fixed points set for a Lotka-Volterra system, Proceedings of Numerical Analysis and Approximation Theory, Cluj–Napoca, Romania, July 5-8, 2006, 331-336.

[109]  D. Otrocol, Differentiability with respect to delays for a Lotka-Volterra system, Creative Math. & Inf., 16 (2007), 36-41.

[110]  C. V. Pao, Global attractor of coupled difference equations and applications to Lotka-Volterra systems, Adv. Difference. Equ., (2005), no. 1, 57-79.

[111]  C. V. Pao, Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion, Nonlinear Analysis, 60 (2005), no. 7, 1197-1217.

[112]  J. H. Park, Stability for a competitive Lotka-Volterra system with delays: LMI optimization approach, Appl. Math. Lett., 18 (2005), no. 6, 689-694.

[113]  L. C. Piccinini, G. Stampachia, G. Vidosich, Ordinary differential equations, Springer Verlag, New York, 1984.

[114]  T. Popoviciu, Analiză numerică, Vol. I, Ed. Acad. RSR, Bucureşti, 1975.

[115]  T. Postelnicu, P. Tăutu, Metode matematice în medicină şi biologie, Ed. Tehnică, Bucureşti, 1971.

[116]  R. Precup, Ecuaţii integrale neliniare, Univ. Babeş-Bolyai, Cluj-Napoca, 1993.

[117]  R. Precup, Monotone technique to the initial values problem for a delay integral equation from Biomathematics, Studia Univ. “Babeş-Bolyai” Math., 40 (1995), no. 2, 63-73.

[118]  R. Redheffer, D. Port, Differential equations theory and applications, Boston, 1991.

[119]  L. W. Roeger, A nonstandard discretization method for Lotka-Volterra models that preserves periodic solutions, J. Difference Equ. Appl., 11 (2005), no. 8, 721-733.

[120]  B. Rus, I. A. Rus, Algebraic properties of the operator A^{∞}, Studia Univ. “Babeş-Bolyai”, 45 (2000), no. 3, 65-68.

[121]  I. A. Rus, Generalized contractions, Seminar of Fixed Point Theory, “Babeş-Bolyai” University, (1983), 1-130.

[122]  I. A. Rus, Basic problems of the metric fixed point theory, I, Studia Univ. “Babeş-Bolyai” Math., 34 (1989), 61-69.

[123]  I. A. Rus, Principii si aplicaţii ale teoriei punctului fix, Editura Dacia, Cluj Napoca, 1979.

[124]  I. A. Rus, Weakly Picard mappings, Comment. Math. Univ. Caroline, 34 (1993), 769–773.

[125]  I. A. Rus, Picard operators and applications, Babeş-Bolyai Univ., Seminar on Fixed Point Theory, 1996.

[126]  I. A. Rus, Ecuaţii diferenţiale. Ecuaţii integrale şi sisteme dinamice, Transilvania Press, Cluj-Napoca, 1996.

[127]  I. A. Rus, Some open problems of fixed point theory, Seminar on Fixed Point Theory (1999), Cluj-Napoca, 19-39

[128]  I. A. Rus, Fiber Picard operators on generalized metric space and applications, Scripta Scientiarum Mathematicarum, 1 (1999), no. 2, 355-363.

[129]  I. A. Rus, Weakly Picard operators and applications, Seminar of Fixed Point Theory, Cluj-Napoca, 2 (2001), 41-58.

[130]  I. A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, 2001.

[131]  I. A. Rus, Functional-differential equation of mixed type, via weakly Picard operators, Seminar of Fixed Point Theory, Cluj-Napoca, 3 (2002), 335-346.

[132]  I. A. Rus, V. Darzu-Ilea, First order functional-differential equations with both advanced and retarded arguments, Seminar of Fixed Point Theory, Cluj-Napoca, 5 (2004), no. 1.

[133]  I. A. Rus, Fixed points, upper and lower fixed points: abstract Gronwall lemmas, Carpathian Journal Math., 20 (2004), no. 1, 125-134.

[134]  I. A. Rus, C. Iancu, Modelare matematică, Casa de editură Transilvania Press, Cluj-Napoca, 2000.

[135]  I. A. Rus, S. Mureşan, Data dependence of the fixed point set of some weakly Picard operators, Seminar Itinerant, Cluj-Napoca (2000), 201-208.

[136]  Y. Saito, T. Hara, W. Ma, Necessary and sufficient conditions for permanence and global stability of a Lotka-Volterra system with two delays, J. Math. Anal. Appl., 236 (1999), 534-556.

[137]  Y. Saito, W. Ma, T. Hara, A necessary and sufficient conditions for permanence of a Lotka-Volterra discrete system with delays, J. Math. Anal. Appl., 256 (2001), 162-174.

[138]  G. M. Scarpellor, D. Ritelli, A new method for the explicit integration of Lotka-Volterra equations, Divulgaciones Mathematicas, 11 (2003), no. 1, 1-17.

[139]  L. F. Shampine, S. Thompson, Solving delay differential equations with dde23, March 2000, http://www.radford.edu/ thompson /webddes.

[140]  L.F. Shampine, S. Thompson, Solving DDEs in MATLAB, Appl. Numer. Math., 37 (2001), 441-458.

[141]  A. Shibata, N. Saito, Time delays and chaos in two competing species, Mathematical Biosciences, 51 (1980), 199-211.

[142]  D. D. Stancu, Gh. Coman, O. Agratini, R. Trîmbiţaş, Analiză numerică şi teoria aproximării, Vol. 1, Presa Universitară Clujeană, Cluj-Napoca, 2001.

[143]  D. D. Stancu, Gh. Coman, P. Blaga, Analiză numerică şi teoria aproximării, Vol. 2, Presa Universitară Clujeană, Cluj-Napoca, 2002.

[144] M. A. Şerban, Fiber ϕ-contractions, Studia Univ. “Babeş-Bolyai”, Mathematica, 44 (1999), no. 3, 99-108.

[145]  M. A. Şerban, Teoria punctului fix pentru operatori definiţi pe produs cartezian, Presa Universitară Clujeană, Cluj-Napoca, 2002.

[146]  Al. Tămăşan, Note on differentiability with respect to delay, P.U.M.A., 9 (1998), no. 1-2 , 215-220.

[147]  B. R. Tang, Y. Kuang, Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional differential systems, Tohoku Mathematical Journal, 49 (1997), 217-239.

[148]  Z. Teng, Nonautonomous Lotka-Volterra systems with delays, Journal of Differential

[149]  D. Trif, Metode numerice pentru ecuaţii diferenţiale şi sisteme dinamice, Casa de Editură Transilvania Press, Cluj-Napoca, 1997.

[150]  R. Trîmbiţaş, Analiza numerică. O introducere bazată pe matlab, Presa Universitară Clujeană, Cluj-Napoca, 2005.

[151]  V. Volterra, V., Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Rendiconti dell’Accademia dei Lincei, 6 (2), (1926), 31–113. Anabridged English version has been published in Fluctuations in the Abundance of a Species Considered Mathematically, Nature 118 (1926), 558-560.

[152]  W. Wang, Z. Ma, Harmless delays for uniform persistence, J. Math. Anal. Appl., 158 (1991), 256-268.

[153]  L. L. Wang, Z. C.Wang, Pozitive periodic solutions of periodic retarded functional differential equations, Applied Mathematics E-Notes, 3 (2003), 138-146.

[154]  H. W. Wu, The existence of positive periodic solution for a neutral delay Lotka-Volterra system, Acta Anal. Funct. Appl., 6 (2004), no. 4, 297-305.

[155]  R. Xu, Global analysis in a pure-delayed type Lotka-Volterra system with one predator and two preys. Contemporary differential equations and applications, 81-90, Nova Sci. Publ.,

[156]  R. Xu, L. Chen, F. Hao, Periodic solutions of an n species Lotka-Volterra type food-chain model with time delays, Appl. Math. Comput., 171 (2005), no. 1, 511-530.

[157]  R. Xu, L. Chen, F. Hao, Periodic solutions of a discrete time Lotka-Volterra type food-chain model with delays, Appl. Math. Comput., 171 (2005), no. 1, 91-103.

[158]  G. Zhang, S. S. Cheng, Positive periodic solutions of coupled delay differential systems depending on two parameters, Taiwanese Journal of Mathematics, 8 (2004), no. 4, 639-652.


Related Posts