 Research
 Open access
 Published:
Dynamics of a discrete LotkaVolterra model
Advances in Difference Equations volume 2013, Article number: 95 (2013)
Abstract
In this paper, we study the equilibrium points, local asymptotic stability of equilibrium points, and global behavior of equilibrium points of a discrete LotkaVolterra model given by
where parameters \alpha ,\beta ,\gamma ,\delta ,\u03f5,\eta \in {\mathbb{R}}^{+}, and initial conditions {x}_{0}, {y}_{0} are positive real numbers. Moreover, the rate of convergence of a solution that converges to the unique positive equilibrium point is discussed. Some numerical examples are given to verify our theoretical results.
MSC:39A10, 40A05.
1 Introduction and preliminaries
Many authors investigated the ecological competition systems governed by differential equations of LotkaVolterra type. Many interesting results related with the global character and local asymptotic stability have been obtained. We refer to [1, 2] and the references therein. Already, many authors [3, 4] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations are of nonoverlapping generations. Particularly, the persistence, boundedness, local asymptotic stability, global character, and the existence of positive periodic solutions.
The discrete LotkaVolterra models have many applications in applied sciences. Such models were first established in mathematical biology, and then their applications were spread to other fields [5–8]. Several variations of the LotkaVolterra predatorprey model have been proposed that offer more realistic descriptions of the interactions of the populations. If the population of rabbits is always much larger than the number of foxes, then the considerations that entered into the development of the logistic equation may come into play. If the number of rabbits becomes sufficiently great, then the rabbits may be interfering with each other in their quest for food and space. One way to describe this effect mathematically is to replace the original model by the more complicated system. Most predators feed on more than one type of food. If the foxes can survive on an alternative resource, although the presence of their natural prey (rabbits) favors growth, a possible alternative model is the discrete dynamical system
where parameters \alpha ,\beta ,\gamma ,\delta ,\u03f5,\eta \in {\mathbb{R}}^{+}, and initial conditions {x}_{0}, {y}_{0} are positive real numbers.
It is a wellknown fact that the discretetime type models described by difference equations are more suitable than the continuoustime models. Nonlinear difference equations of order greater than one are of paramount importance in applications. Such equations also appear naturally as discrete analogues and as numerical solutions of differential and delay differential equations which model various diverse phenomena in biology, ecology, physiology, physics, engineering, and economics. Rational difference equations are a special form of nonlinear difference equations. We refer to [9–14] for basic theory of difference equations and rational difference equations. Recently, many authors have discussed the dynamics of rational difference equations [15–27].
2 Linearized stability
Let us consider a twodimensional discrete dynamical system of the form
where f:I\times J\to I and g:I\times J\to J are continuously differentiable functions and I, J are some intervals of real numbers. Furthermore, a solution {\{({x}_{n},{y}_{n})\}}_{n=0}^{\mathrm{\infty}} of the system (2.1) is uniquely determined by initial conditions ({x}_{0},{y}_{0})\in I\times J. An equilibrium point of (2.1) is a point (\overline{x},\overline{y}) that satisfies
Definition 2.1 Let (\overline{x},\overline{y}) be an equilibrium point of the system (2.1).

(i)
An equilibrium point (\overline{x},\overline{y}) is said to be stable if for every \epsilon >0 there exists \delta >0 such that for every initial condition ({x}_{0},{y}_{0}) if \parallel ({x}_{0},{y}_{0})(\overline{x},\overline{y})\parallel <\delta implies \parallel ({x}_{n},{y}_{n})(\overline{x},\overline{y})\parallel <\epsilon for all n>0, where \parallel \cdot \parallel is the usual Euclidean norm in {\mathbb{R}}^{2}.

(ii)
An equilibrium point (\overline{x},\overline{y}) is said to be unstable if it is not stable.

(iii)
An equilibrium point (\overline{x},\overline{y}) is said to be asymptotically stable if there exists \eta >0 such that \parallel ({x}_{0},{y}_{0})(\overline{x},\overline{y})\parallel <\eta and ({x}_{n},{y}_{n})\to (\overline{x},\overline{y}) as n\to \mathrm{\infty}.

(iv)
An equilibrium point (\overline{x},\overline{y}) is called a global attractor if ({x}_{n},{y}_{n})\to (\overline{x},\overline{y}) as n\to \mathrm{\infty}.

(v)
An equilibrium point (\overline{x},\overline{y}) is called an asymptotic global attractor if it is a global attractor and stable.
Definition 2.2 Let (\overline{x},\overline{y}) be an equilibrium point of a map F(x,y)=(f(x,y),g(x,y)), where f and g are continuously differentiable functions at (\overline{x},\overline{y}). The linearized system of (2.1) about the equilibrium point (\overline{x},\overline{y}) is given by
where {X}_{n}=\left(\begin{array}{c}{x}_{n}\\ {y}_{n}\end{array}\right) and {F}_{J} is a Jacobian matrix of the system (2.1) about the equilibrium point (\overline{x},\overline{y}).
Let (\overline{x},\overline{y}) be an equilibrium point of the system (1.1), then
Hence, O=(0,0), P=(\frac{\beta \beta \delta +(1+\alpha )\eta}{\beta \u03f5+\gamma \eta},\frac{\gamma (1+\delta )+(1+\alpha )\u03f5}{\beta \u03f5+\gamma \eta}), Q=(\frac{1+\alpha}{\gamma},0), and R=(0,\frac{1+\delta}{\eta}) are equilibrium points of the system (1.1). Then, clearly, P=(\frac{\beta \beta \delta +(1+\alpha )\eta}{\beta \u03f5+\gamma \eta},\frac{\gamma (1+\delta )+(1+\alpha )\u03f5}{\beta \u03f5+\gamma \eta}) is the unique positive equilibrium point of the system (1.1), if \alpha >1, \delta \le 1, \u03f5>\frac{\gamma \gamma \delta}{\alpha 1} or \alpha >1, \delta >1, \eta >\frac{\beta +\beta \delta}{1+\alpha}.
The Jacobian matrix of the linearized system of (1.1) about the fixed point (\overline{x},\overline{y}) is given by
Theorem 2.3 For the system {X}_{n+1}=F({X}_{n}), n=0,1,\dots , of difference equations such that \overline{X} is a fixed point of F. If all eigenvalues of the Jacobian matrix {J}_{F} about \overline{X} lie inside the open unit disk \lambda <1, then \overline{X} is locally asymptotically stable. If one of them has a modulus greater than one, then \overline{X} is unstable.
3 Main results
Theorem 3.1 Assume that \alpha <1 and \delta <1, then the following statements are true.

(i)
The equilibrium point O=(0,0) is locally asymptotically stable.

(ii)
The equilibrium point Q=(\frac{1+\alpha}{\gamma},0) is unstable.

(iii)
The equilibrium point R=(0,\frac{1+\delta}{\eta}) is unstable.
Proof (i) The Jacobian matrix of the linearized system of (1.1) about the fixed point (0,0) is given by
Moreover, the eigenvalues of the Jacobian matrix {J}_{F}(0,0) about (0,0) are {\lambda}_{1}=\alpha <1 and {\lambda}_{2}=\delta <1. Hence, the equilibrium point (0,0) is locally asymptotically stable.

(ii)
The Jacobian matrix of the linearized system of (1.1) about the fixed point (\frac{1+\alpha}{\gamma},0) is given by
{F}_{J}(\frac{1+\alpha}{\gamma},0)=\left[\begin{array}{cc}\frac{1}{\alpha}& \frac{\beta \alpha \beta}{\alpha \gamma}\\ 0& \frac{(\alpha 1)\u03f5+\delta \gamma}{\gamma}\end{array}\right].
The eigenvalues of the Jacobian matrix {J}_{F}(\frac{1+\alpha}{\gamma},0) about (\frac{1+\alpha}{\gamma},0) are {\lambda}_{1}=\frac{1}{\alpha}>1 and {\lambda}_{2}=\frac{\gamma \delta \u03f5+\alpha \u03f5}{\gamma}.

(iii)
The Jacobian matrix of the linearized system of (1.1) about the fixed point (0,\frac{1+\delta}{\eta}) is given by
{F}_{J}(0,\frac{1+\delta}{\eta})=\left[\begin{array}{cc}\frac{\beta \beta \delta +\alpha \eta}{\eta}& 0\\ \frac{(1+\delta )\u03f5}{\delta \eta}& \frac{1}{\delta}\end{array}\right].
The eigenvalues of the Jacobian matrix {J}_{F}(0,\frac{1+\delta}{\eta}) about (0,\frac{1+\delta}{\eta}) are {\lambda}_{1}=\frac{1}{\delta}>1 and {\lambda}_{2}=\frac{\beta \beta \delta +\alpha \eta}{\eta}. □
Theorem 3.2 The following statements are true.

(i)
If \alpha >1, \delta <1, and \u03f5<\frac{\gamma \gamma \delta}{\alpha 1}, then the equilibrium point Q=(\frac{1+\alpha}{\gamma},0) is locally asymptotically stable.

(ii)
If \delta >1 and \alpha <1, then the equilibrium point R=(0,\frac{1+\delta}{\eta}) is locally asymptotically stable.
Theorem 3.3 Assume that \alpha >1, \delta >1, and \eta >\frac{\beta +\beta \delta}{1+\alpha}, then the unique equilibrium point P=(\frac{\beta \beta \delta +(1+\alpha )\eta}{\beta \u03f5+\gamma \eta},\frac{\gamma (1+\delta )+(1+\alpha )\u03f5}{\beta \u03f5+\gamma \eta}) is locally asymptotically stable if
where
Proof Assume that \alpha >1, \delta >1, and \eta >\frac{\beta +\beta \delta}{1+\alpha}. Let L=\beta \beta \delta +(1+\alpha )\eta >0. Then a characteristic polynomial of the Jacobian matrix {F}_{J}(P) about the unique equilibrium point P=(\frac{L}{\beta \u03f5+\gamma \eta},\frac{\gamma (1+\delta )+(1+\alpha )\u03f5}{\beta \u03f5+\gamma \eta}) is given by
where
and
Let
Assume that \mathrm{\Omega}<{(\beta (\gamma \gamma \delta +\u03f5)+\alpha \gamma \eta )}^{2}(\gamma \delta \eta +\u03f5(\beta +(1+\alpha )\eta )). Then one has
Then, by Rouche’s theorem, S(\lambda ) and S(\lambda )T(\lambda ) have the same number of zeroes in an open unit disk \lambda <1. Hence, the unique positive equilibrium point P is locally asymptotically stable. □
3.1 Global character
Theorem 3.4 Let I=[a,b] and J=[c,d] be real intervals, and let f:I\times J\to I and g:I\times J\to J be continuous functions. Consider the system (2.1) with initial conditions ({x}_{0},{y}_{0})\in I\times J. Suppose that the following statements are true.

(i)
f(x,y) is nondecreasing in x and nonincreasing in y.

(ii)
g(x,y) is nondecreasing in both arguments.

(iii)
If ({m}_{1},{M}_{1},{m}_{2},{M}_{2})\in {I}^{2}\times {J}^{2} is a solution of the system
\begin{array}{c}{m}_{1}=f({m}_{1},{M}_{2}),\phantom{\rule{2em}{0ex}}{M}_{1}=f({M}_{1},{m}_{2}),\hfill \\ {m}_{2}=g({m}_{1},{m}_{2}),\phantom{\rule{2em}{0ex}}{M}_{2}=g({M}_{1},{M}_{2})\hfill \end{array}
such that {m}_{1}={M}_{1} and {m}_{2}={M}_{2}, then there exists exactly one equilibrium point (\overline{x},\overline{y}) of the system (2.1) such that {lim}_{n\to \mathrm{\infty}}({x}_{n},{y}_{n})=(\overline{x},\overline{y}).
Proof According to the Brouwer fixed point theorem, the function F:I\times J\to I\times J defined by F(x,y)=F(f(x,y),g(x,y)) has a fixed point (\overline{x},\overline{y}), which is a fixed point of the system (2.1).
Assume that {m}_{1}^{0}=a, {M}_{1}^{0}=b, {m}_{2}^{0}=c, {M}_{2}^{0}=d such that
and
Then
and
Moreover, one has
and
We similarly have
and
Now observe that for each i\ge 0,
and
Hence, {m}_{1}^{i}\le {x}_{n}\le {M}_{1}^{i}, and {m}_{2}^{i}\le {y}_{n}\le {M}_{2}^{i} for n\ge 2i+1. Let {m}_{1}={lim}_{n\to \mathrm{\infty}}{m}_{1}^{i}, {M}_{1}={lim}_{n\to \mathrm{\infty}}{M}_{1}^{i}, {m}_{2}={lim}_{n\to \mathrm{\infty}}{m}_{2}^{i}, and {M}_{2}={lim}_{n\to \mathrm{\infty}}{M}_{2}^{i}. Then a\le {m}_{1}\le {M}_{1}\le b and c\le {m}_{2}\le {M}_{2}\le d. By the continuity of f and g, one has
Hence, {m}_{1}={M}_{1}, {m}_{2}={M}_{2}. □
Theorem 3.5 Assume that \eta \gamma \beta \u03f5\ne 0, then the unique positive equilibrium point P of the system (1.1) is a global attractor.
Proof Let f(x,y)=\frac{\alpha x\beta xy}{1+\gamma x} and g(x,y)=\frac{\delta y+\u03f5xy}{1+\eta y}. Then it is easy to see that f(x,y) is nondecreasing in x and nonincreasing in y. Moreover, g(x,y) is nondecreasing in both x and y. Let ({m}_{1},{M}_{1},{m}_{2},{M}_{2}) be a positive solution of the system
Then one has
and
From (3.1), one has
On subtraction, (3.3) implies that
Similarly, from (3.2), one has
On subtraction, (3.5) implies that
Comparing (3.4) and (3.6), one has
Then one has {m}_{1}={M}_{1} and {m}_{2}={M}_{2}. Hence, from Theorem 3.4 the equilibrium point (\frac{\beta \beta \delta +(1+\alpha )\eta}{\beta \u03f5+\gamma \eta},\frac{\gamma (1+\delta )+(1+\alpha )\u03f5}{\beta \u03f5+\gamma \eta}) of the system (1.1) is a global attractor. □
Theorem 3.6 Assume that \alpha >1, \delta >1, and \eta \gamma \beta \u03f5\ne 0. Then the unique positive equilibrium point (\overline{x},\overline{y})=(\frac{\beta \beta \delta +(1+\alpha )\eta}{\beta \u03f5+\gamma \eta},\frac{\gamma (1+\delta )+(1+\alpha )\u03f5}{\beta \u03f5+\gamma \eta}) is globally asymptotically stable.
Proof The proof follows from Theorem 3.3 and Theorem 3.5. □
3.2 Rate of convergence
In this section we determine the rate of convergence of a solution that converges to the unique positive equilibrium point of the system (1.1).
The following result gives the rate of convergence of solutions of a system of difference equations:
where {X}_{n} is an mdimensional vector, A\in {C}^{m\times m} is a constant matrix, and B:{\mathbb{Z}}^{+}\to {C}^{m\times m} is a matrix function satisfying
as n\to \mathrm{\infty}, where \parallel \cdot \parallel denotes any matrix norm which is associated with the vector norm
Proposition 3.7 (Perron’s theorem [28])
Suppose that condition (3.8) holds. If {X}_{n} is a solution of (3.7), then either {X}_{n}=0 for all large n or
exists and is equal to the modulus of one of the eigenvalues of matrix A.
Proposition 3.8 [28]
Suppose that condition (3.8) holds. If {X}_{n} is a solution of (3.7), then either {X}_{n}=0 for all large n or
exists and is equal to the modulus of one of the eigenvalues of matrix A.
Let \{({x}_{n},{y}_{n})\} be any solution of the system (1.1) such that {lim}_{n\to \mathrm{\infty}}{x}_{n}=\overline{x}, and {lim}_{n\to \mathrm{\infty}}{y}_{n}=\overline{y}, where (\overline{x},\overline{y})=(\frac{\beta \beta \delta +(1+\alpha )\eta}{\beta \u03f5+\gamma \eta},\frac{\gamma (1+\delta )+(1+\alpha )\u03f5}{\beta \u03f5+\gamma \eta}). To find the error terms, one has from the system (1.1)
and
Let {e}_{n}^{1}={x}_{n}\overline{x} and {e}_{n}^{2}={y}_{n}\overline{y}, then one has
and
where
Moreover,
Now the limiting system of error terms can be written as
which is similar to the linearized system of (1.1) about the equilibrium point (\overline{x},\overline{y}).
Using Proposition 3.7, one has following result.
Theorem 3.9 Assume that \{({x}_{n},{y}_{n})\} is a positive solution of the system (1.1) such that {lim}_{n\to \mathrm{\infty}}{x}_{n}=\overline{x} and {lim}_{n\to \mathrm{\infty}}{y}_{n}=\overline{y}, where
Then the error vector {e}_{n}=\left(\begin{array}{c}{e}_{n}^{1}\\ {e}_{n}^{2}\end{array}\right) of every solution of (1.1) satisfies both of the following asymptotic relations:
where {\lambda}_{1,2}{F}_{J}(\overline{x},\overline{y}) are the characteristic roots of the Jacobian matrix {F}_{J}(\overline{x},\overline{y}).
4 Examples
In this section, we consider some numerical examples which show that under a suitable choice of parameters α, β, γ, δ, ϵ, η, the unique positive equilibrium point (\frac{\beta \beta \delta +(1+\alpha )\eta}{\beta \u03f5+\gamma \eta},\frac{\gamma (1+\delta )+(1+\alpha )\u03f5}{\beta \u03f5+\gamma \eta}) of the system (1.1) is globally asymptotically stable.
Example Let \alpha =1.001, \beta =0.03, \gamma =0.6, \delta =1.002, \u03f5=1.7, \eta =0.9. Then the system (1.1) can be written as
with initial conditions {x}_{0}=0.0002, {y}_{0}=0.0006.
In this case, the unique positive equilibrium point P of the system (4.1) is given by
Moreover, the plot is shown in Figure 1.
Example Let \alpha =2.5, \beta =0.7, \gamma =2.3, \delta =2.7, \u03f5=4.2, \eta =6.8. Then the system (1.1) can be written as
with initial conditions {x}_{0}=0.84, {y}_{0}=0.5.
In this case, the unique equilibrium point P of the system (4.2) is given by
Moreover, the plot is shown in Figure 2.
Example Let \alpha =22, \beta =1.7, \gamma =20.5, \delta =6, \u03f5=0.2, \eta =2.8. Then the system (1.1) can be written as
with initial conditions {x}_{0}=0.7, {y}_{0}=0.8.
In this case, the unique equilibrium point P of the system (4.3) is given by
Moreover, the plot is shown in Figure 3.
Example Let \alpha =170, \beta =11, \gamma =2.7, \delta =50, \u03f5=1.7, \eta =7. Then the system (1.1) can be written as
with initial conditions {x}_{0}=7, {y}_{0}=5.
In this case, the unique positive equilibrium point P of the system (4.4) is given by
Moreover, the plot of the system (4.4) is shown in Figure 4. An attractor of the system is shown in Figure 5.
5 Conclusions
This work is related to the qualitative behavior of a discretetime LotkaVolterra model. The continuous form of this model is given by
where a, b, c, m, n, p are positive constants. Moreover, the discrete form (1.1) of the continuous model is obtained by using some nonstandard difference scheme such that the equilibrium points in both cases are conserved. We proved that the system (1.1) has four equilibrium points, which are locally asymptotically stable under certain conditions. The main contribution in this paper is to prove that the unique positive equilibrium point
of the system (1.1) is globally asymptotically stable. Furthermore, we have investigated the rate of convergence of the solution that converges to the unique positive equilibrium point of the system (1.1). Some numerical examples are provided to support our theoretical results. These examples are experimental verifications of theoretical discussions.
Author’s contributions
The author carried out the proof of the main results and approved the final manuscript.
References
Ahmad S: On the nonautonomous LotkaVolterra competition equation. Proc. Am. Math. Soc. 1993, 117: 199–204. 10.1090/S00029939199311430133
Tang X, Zou X: On positive periodic solutions of LotkaVolterra competition systems with deviating arguments. Proc. Am. Math. Soc. 2006, 134: 2967–2974. 10.1090/S0002993906083201
Zhou Z, Zou X: Stable periodic solutions in a discrete periodic logistic equation. Appl. Math. Lett. 2003, 16(2):165–171. 10.1016/S08939659(03)800277
Liu X: A note on the existence of periodic solution in discrete predatorprey models. Appl. Math. Model. 2010, 34: 2477–2483. 10.1016/j.apm.2009.11.012
Krebs W: A general predatorprey model. Math. Comput. Model. Dyn. Syst. 2003, 9: 387–401. 10.1076/mcmd.9.4.387.27896
Allen LJS: An Introduction to Mathematical Biology. Prentice Hall, New York; 2007.
Brauer F, CastilloChavez C: Mathematical Models in Population Biology and Epidemiology. Springer, Berlin; 2000.
EdelsteinKeshet L: Mathematical Models in Biology. McGrawHill, New York; 1988.
Kulenović MRS, Ladas G: Dynamics of Second Order Rational Difference Equations. Chapman & Hall/CRC Press, Boca Raton; 2002.
Camouzis E, Ladas G: Dynamics of ThirdOrder Rational Difference Equations with Open Problems and Conjectures. Chapman & Hall/CRC Press, Boca Raton; 2007.
Elaydi S: An Introduction to Difference Equations. 3rd edition. Springer, New York; 2005.
Agarwal RP, Wong P: Advanced Topics in Difference Equations. Kluwer Academic, Dordrecht; 1997.
Agarwal RP: Difference Equations and Inequalities. 1st edition. Dekker, New York; 1992. (2nd edn. (2000))
Grove EA, Ladas G: Periodicities in Nonlinear Difference Equations. Chapman & Hall/CRC Press, Boca Raton; 2004.
Aloqeili M: Dynamics of a rational difference equation. Appl. Math. Comput. 2006, 176(2):768–776. 10.1016/j.amc.2005.10.024
Cinar C:On the positive solutions of the difference equation system {x}_{n+1}=\frac{1}{{y}_{n}}; {y}_{n+1}=\frac{{y}_{n}}{{x}_{n1}{y}_{n1}}. Appl. Math. Comput. 2004, 158: 303–305. 10.1016/j.amc.2003.08.073
Stević S: On some solvable systems of difference equations. Appl. Math. Comput. 2012, 218: 5010–5018. 10.1016/j.amc.2011.10.068
Bajo I, Liz E: Global behaviour of a secondorder nonlinear difference equation. J. Differ. Equ. Appl. 2011, 17(10):1471–1486. 10.1080/10236191003639475
Kalabuŝić S, Kulenović MRS, Pilav E: Dynamics of a twodimensional system of rational difference equations of LeslieGower type. Adv. Differ. Equ. 2011. doi:10.1186/1687–1847–2011–29
Touafek N, Elsayed EM: On the solutions of systems of rational difference equations. Math. Comput. Model. 2012, 55: 1987–1997. 10.1016/j.mcm.2011.11.058
Touafek N, Elsayed EM: On the periodicity of some systems of nonlinear difference equations. Bull. Math. Soc. Sci. Math. Roum. 2012, 2: 217–224.
Din, Q: On a system of rational difference equation. Demonstr. Math. (in press)
Din, Q: Global character of a rational difference equation. Thai J. Math. (in press)
Din Q, Qureshi MN, Khan AQ: Dynamics of a fourthorder system of rational difference equations. Adv. Differ. Equ. 2012. doi:10.1186/1687–1847–2012–216
Zhang Q, Yang L, Liu J: Dynamics of a system of rational third order difference equation. Adv. Differ. Equ. 2012. doi:10.1186/1687–1847–2012–136
Shojaei M, Saadati R, Adibi H: Stability and periodic character of a rational third order difference equation. Chaos Solitons Fractals 2009, 39: 1203–1209. 10.1016/j.chaos.2007.06.029
Elsayed EM: Behavior and expression of the solutions of some rational difference equations. J. Comput. Anal. Appl. 2013, 15(1):73–81.
Pituk M: More on Poincare’s and Perron’s theorems for difference equations. J. Differ. Equ. Appl. 2002, 8: 201–216. 10.1080/10236190211954
Acknowledgements
The author would like to thank the main editor and anonymous referees for their valuable comments and suggestions leading to improvement of this paper. This work was supported by the Higher Education Commission of Pakistan.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Din, Q. Dynamics of a discrete LotkaVolterra model. Adv Differ Equ 2013, 95 (2013). https://doi.org/10.1186/16871847201395
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/16871847201395