# On stochastic Lotka-Volterra predator-prey model We are searching data for your request:

Forums and discussions:
Manuals and reference books:
Data from registers:
Wait the end of the search in all databases.
Upon completion, a link will appear to access the found materials.

I am currently working (as a mathematician) on some estimations involving an stochastic predator-prey type model in which some of the coefficients have been perturbed by a Brownian Motion yielding to a system of stochastic differential equations.

The basic formulation of this problem can be found for instance in this article.

We have the following model

$$egin{equation}label{0} egin{cases} dX(t)=X(t)left(a-bX(t)-frac{sY(t)}{eta+Y(t)} ight)dt X(0)=x>0, dY(t)=Y(t)left(frac{h X(t)}{eta+Y(t)}-c-fY(t) ight)dt Y(0)=y>0. end{cases} end{equation}$$

where we use Holling II response function.

In the literature (mostly mathematics literature) the authors propose to perturb the birth rate of the preys $$amapsto a+dot{B}_1(t)$$ and the mortality rate of predators $$cmapsto c+dot{B}_2(t)$$.

This yields to the following (stochastic) system

$$egin{equation}label{1} egin{cases} dX(t)=X(t)left(a-bX(t)-frac{sY(t)}{eta+Y(t)} ight)dt+sigma_1 X(t)dB_1(t), X(0)=x>0, dY(t)=Y(t)left(frac{h X(t)}{eta+Y(t)}-c-fY(t) ight)dt+sigma_2 Y(t)dB_2(t), Y(0)=y>0 end{cases} end{equation}$$

I know nothing about biology but I am concerned about a couple of things regarding this particular formulation:

1. If we assume that the Brownian motions are perturbing the birth/mortality rates, how can I interpret the fact that this new "perturbed" rates can become negative (due to the effect of the BM). Does it have any sense (in the framework of this particular model) to talk about "negative birth rate" or "negative mortality rate"?
2. What if the "perturbed" mortality rate goes beyond $$100 \%$$? Mathematically nothing "wrong" will happen, but what about the interpretation?

Notice that the "noise" and the model as a whole can be interpreted differently (ignoring the fact that we are actually modelling the dynamics of two species) but my main issue is that if I(together with many authors) am stating that we perturb a certain parameter I believe that we must respect the basic assumptions of the model!

I hope everything's clear and I thank you all in advance, any opinion or suggestion will be welcome!

1. To interpret $$a$$ and $$c$$ as a birth and mortality rates is somewhat inaccurate, as $$a$$ and $$c$$, as written are density-independent rates that grow or shrink populations exponentially. If $$a$$ were negative, the prey population would shrink and most people would find a problem like that unexciting unless it was, say, a sink population maintained by immigration from another population. This is true of $$c$$ being negative, where the predator population would grow independent of the prey population. More accurately, $$a$$ can be interpreted as the difference between birth, $$b$$, and deaths, $$delta$$; i.e., $$a = b - delta$$. In this case, depending on the kinds of stochasticity being added to the model, negative $$a$$ is not just normal, but expected. The same is true for $$c$$. Ultimately, a perturbations like the white noise in this equation is simultaneously perturbing the difference between birth and death rates, which explains the negative difference.
2. The value of the rates seem to have the same interpretation if they are constant or are perturbed to an arbitrary magnitude.

I hope this helps!

## On stochastic Lotka-Volterra predator-prey model - Biology

We investigate spatially inhomogeneous versions of the stochastic Lotka-Volterra model for predator-prey competition and coexistence by means of Monte Carlo simulations on a two-dimensional lattice with periodic boundary conditions. To study boundary effects for this paradigmatic population dynamics system, we employ a simulation domain split into two patches: Upon setting the predation rates at two distinct values, one half of the system resides in an absorbing state where only the prey survives, while the other half attains a stable coexistence state wherein both species remain active. At the domain boundary, we observe a marked enhancement of the predator population density. The predator correlation length displays a minimum at the boundary, before reaching its asymptotic constant value deep in the active region. The frequency of the population oscillations appears only very weakly affected by the existence of two distinct domains, in contrast to their attenuation rate, which assumes its largest value there. We also observe that boundary effects become less prominent as the system is successively divided into subdomains in a checkerboard pattern, with two different reaction rates assigned to neighboring patches. When the domain size becomes reduced to the scale of the correlation length, the mean population densities attain values that are very similar to those in a disordered system with randomly assigned reaction rates drawn from a bimodal distribution.

Abundo, M., Rossi C.: Numerical simulation of a stochastic model for cancerous cells submitted to chemotherapy. J. Math. Biol. 27, 81–90 (1989)

Andersen, P. K., Borgan, Ø.: Counting process models for life history data. Scand. J. Statist. 12, 97–158 (1985)

Arnold, L.: Stochastic differential equations: theory and applications. New York: Wiley 1974

Arnold, L., Lefever, R. (eds.) Stochastic non linear systems in physics, chemistry and biology. (Springer Series in Synergetics) Berlin Heidelberg New York: Springer 1981

Barra, M., Del Grosso, G., Gerardi, A., Koch, G., Marchetti, F.: Some basic properties of stochastic population models. (Lect. Notes Biomath., vol. 32, pp. 155–164) Berlin Heidelberg New York: Springer 1978

Billard, L. On Lotka-Volterra predator-prey models. J. Appl. Probab. 14, 375–381 (1977)

Billingsley, P.: The Lindeberg-Levy theorem for martingales. Proc. Am. Math. Soc. 12, 788, 792 (1961)

Billingsley, P.: Convergence of probability measures. New York: Wiley 1968

Capocelli, R. M., Ricciardi, L. M.: A diffusion model for population growth in random environment. Theor. Popul. Biol. 5, 28–45 (1974)

Crow, J. F., Kimura, M.: An introduction to population genetics theory. New York: Harper and Row 1970

Feldman, M. W., Roughgarden, J.: A population's stationary distribution and chance of extinction with remarks on the theory of species packing. Theor. Popul. Biol. 7, 197–207 (1975)

Ferrante, L., Koch, G.: An application of Liapunov techniques to stochastic population models. Report Dipartimento di Matematica University di Roma, ‘La Sapienza’

Gihman, I. I., Skorohod, A. V.: Stochastic differential equations. Berlin Heidelberg New York: Springer 1972

Goel, N. S., Maitra, S. C., Montroll, E. W.: On the Volterra and other nonlinear models of interacting populations. Rev. Modern Phys. 43, 241–276 (1971)

Hinkley, S. W., Tsokos, C. P.: A stochastic model for chemical equilibrium. Math. Biol. 21, 241–276 (1971)

Khas'minskiy, R. Z.: Stability of systems of differential equations in the presence of random disturbances (in Russian). Moscow: Nauka Press 1969

Koch, G.: Stochastic models in biology I, II. Systems Anal. Modelling Simulation 1, 27–33 151–168 (1984)

Kushner, H. J. Stability and existence of diffusions with discontinuous or rapidly growing drift terms. J. Differ 11, 156–168 (1972)

Lewontin, R. C., Cohen D.: On population growth in randomly varying environment. Proc. Nat. Acad. Sci. 62, 1056–1060 (1969)

Ludwig, D.: Persistence of dynamical systems under random perturbations. SIAM Rev. 15, 605–640 (1975)

Nobile, A. G., Ricciardi, L. M.: Growth and extinction in random environment. (Second International Conference on Information Science and Systems, Univ. Patras, Patras, 1979), vol. III, pp. 455–465. Dordrecht: Reidel 1980

May, R. M.: Stability and complexity in model ecosystems. Princeton University Press 1973

Rescigno, A., Richardson, I. W.: The deterministic theory of population dynamics. In: Rosen R. (ed.) Foundation of mathematical biology, vol. 3. New York: Academic Press 1981

Ricciardi, L. M.: Diffusion process and related topics in biology. (Lect. Notes Biomath., vol. 14) Berlin Heidelberg New York: Springer 1977

Ricciardi, L. M.: Stochastic equations in neurobiology and population biology. (Lect. Notes Biomath., vol. 39, 248–263) Springer Verlag (1980)

Rumelin, W.: Numerical treatment of stochastic differential equations. SIAM J. Numer. Anal. 19, 604–613 (1982)

## 2. Stochastic Differential Equation with Markovian Switching

Throughout this paper, unless otherwise specified, we let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets.) Let , , be a right-continuous Markov chain in the probability space tasking values in a finite state space with generator given by

where . Here, is the transition rate from to if , while . We assume that the Markov chain is independent of the Brownian motion. And almost every sample path of is a right-continuous step function with a finite number of simple jumps in any finite subinterval of .

We assume, as a standing hypothesis in following of the paper, that the Markov chain is irreducible. The algebraic interpretation of irreducibility is rank. Under this condition, the Markov chain has a unique stationary distribution which can be determined by solving the following linear equation:

Department of Mathematical Sciences, College of Science, UAE University, Al Ain, 15551, UAE

* Corresponding author: F.A. Rihan ([email protected])

Received April 2020 Revised September 2020 Published November 2020

Fund Project: This work supported by UPAR-Project (Code # G00003440)

Environmental factors and random variation have strong effects on the dynamics of biological and ecological systems. In this paper, we propose a stochastic delay differential model of two-prey, one-predator system with cooperation among prey species against predator. The model has a global positive solution. Sufficient conditions of existence and uniqueness of an ergodic stationary distribution of the positive solution are provided, by constructing suitable Lyapunov functionals. Sufficient conditions for possible extinction of the predator populations are also obtained. The conditions are expressed in terms of a threshold parameter \$ _0^s \$ that relies on the environmental noise. Illustrative examples and numerical simulations, using Milstein's scheme, are carried out to illustrate the theoretical results. A small scale of noise can promote survival of the species. While relative large noises can lead to possible extinction of the species in such an environment.

##### References:

J. Alebraheem and Y. A. Hasan, Dynamics of a two predator–one prey system, Computational and Applied Mathematics, 33 (2014), 767-780. doi: 10.1007/s40314-013-0093-8. Google Scholar

A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 292 (2004), 364-380. doi: 10.1016/j.jmaa.2003.12.004. Google Scholar

Y. Bai and Y. Li, Stability and Hopf bifurcation for a stage-structured predator–prey model incorporating refuge for prey and additional food for predator, Advances in Difference Equations, 2019 (2019), 1-20. doi: 10.1186/s13662-019-1979-6. Google Scholar

J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, Journal of Mathematical Analysis and Applications, 391 (2012), 363-375. doi: 10.1016/j.jmaa.2012.02.043. Google Scholar

G. A. Bocharov and F. A. Rihan, Numerical modelling in biosciences using delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 183-199. doi: 10.1016/S0377-0427(00)00468-4. Google Scholar

E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 297-307. doi: 10.1016/S0377-0427(00)00475-1. Google Scholar

B. Dubey and A. Kumar, Dynamics of prey–predator model with stage structure in prey including maturation and gestation delays, Nonlinear Dynamics, 96 (2019), 2653-2679. doi: 10.1007/s11071-019-04951-5. Google Scholar

M. F. Elettreby, Two-prey one-predator model, Chaos, Solitons & Fractals, 39 (2009), 2018-2027. doi: 10.1016/j.chaos.2007.06.058. Google Scholar

T. C. Gard, Persistence in stochastic food web models, Bulletin of Mathematical Biology, 46 (1984), 357-370. doi: 10.1016/S0092-8240(84)80044-0. Google Scholar

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. Google Scholar

R. Khasminskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. Google Scholar

P. E. Kloeden and T. Shardlow, The Milstein scheme for stochastic delay differential equations without using anticipative calculus, Stochastic Analysis and Applications, 30 (2012), 181-202. doi: 10.1080/07362994.2012.628907. Google Scholar

S. Kundu and S. Maitra, Dynamical behaviour of a delayed three species predator–prey model with cooperation among the prey species, Nonlinear Dynamics, 92 (2018), 627-643. doi: 10.1007/s11071-018-4079-3. Google Scholar

D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, Journal of Differential Equations, 263 (2017), 8873-8915. doi: 10.1016/j.jde.2017.08.066. Google Scholar

R. S. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228. doi: 10.1080/17442508008833146. Google Scholar

Z. Liu and R. Tan, Impulsive harvesting and stocking in a monod–haldane functional response predator–prey system, Chaos, Solitons & Fractals, 34 (2007), 454-464. doi: 10.1016/j.chaos.2006.03.054. Google Scholar

Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Stationary distribution and extinction of a stochastic predator–prey model with herd behavior, Journal of the Franklin Institute, 355 (2018), 8177-8193. doi: 10.1016/j.jfranklin.2018.09.013. Google Scholar

A. J. Lotka, Elements of Physical Biology, Baltimore: Williams & Wilkins Co., 1925. Google Scholar

X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2008. doi: 10.1533/9780857099402. Google Scholar

X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Processes and their Applications, 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0. Google Scholar

X. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0. Google Scholar

J. D. Murray, Mathematical Biology, Springer New york, 1993. doi: 10.1007/b98869. Google Scholar

R. Rakkiyappan, A. Chandrasekar, F. A. Rihan and S. Lakshmanan, Exponential state estimation of Markovian jumping genetic regulatory networks with mode-dependent probabilistic time-varying delays, Mathematical Biosciences, 25 (2014), 30-53. doi: 10.1016/j.mbs.2014.02.008. Google Scholar

R Rakkiyappan, G. Velmurugan, F. A. Rihan and and S. Lakshmanan, Stability analysis of memristor-based complex-valued recurrent neural networks with time delays, Complexity, 21 (2015), 14-39. doi: 10.1002/cplx.21618. Google Scholar

F. A. Rihan, S. Lakshmanan, A. H. Hashish, R. Rakkiyappan and E. Ahmed, Fractional-order delayed predator–prey systems with functional response, Nonlinear Dynamics, 80 (2015), 777-789. doi: 10.1007/s11071-015-1905-8. Google Scholar

F. A. Rihan, H. J. Alsakaji and C. Rajivganthi, Stability and Hopf bifurcation of three-species prey-predator system with time delays and Allee effect, Complexity, 2020 (2020), 7306412. doi: 10.1155/2020/7306412. Google Scholar

F. A. Rihan, A. A. Azamov and H. J. Al-Sakaji, An inverse problem for delay differential equations: Parameter estimation, nonlinearity, sensitivity, Applied Mathematics & Information Sciences, 12 (2018), 63-74. doi: 10.18576/amis/120106. Google Scholar

F. A. Rihan and H. J. Alsakaji, Persistence and extinction for stochastic delay differential model of prey-predator system with hunting cooperation in predators, Advances in Difference Equations, 124 (2020), 1-22. doi: 10.1186/s13662-020-02579-z. Google Scholar

F.A. Rihan, H.J. Alsakaji and C. Rajivganthi, Stochastic SIRC epidemic model with time-delay for COVID-19, Adv. Differ. Equ., 2020 (2020), 1-20. doi: 10.1186/s13662-020-02964-8. Google Scholar

F. A. Rihan, C. Rajivganthi and P. Muthukumar, Fractional stochastic differential equations with Hilfer fractional derivative: Poisson jumps and optimal control, Discrete Dynamics in Nature and Society, 2017 (2017), Art. ID 5394528, 11 pp. doi: 10.1155/2017/5394528. Google Scholar

T. Saha and M. Bandyopadhyay, Dynamical analysis of a delayed ratio-dependent prey–predator model within fluctuating environment, Applied Mathematics and Computation, 196 (2008), 458-478. doi: 10.1016/j.amc.2007.06.017. Google Scholar

G. Tang, S. Tang and R. A. Cheke, Global analysis of a holling type II predator–prey model with a constant prey refuge, Nonlinear Dynamics, 76 (2014), 635-647. doi: 10.1007/s11071-013-1157-4. Google Scholar

D. A. Vasseuri and P. Yodzis, The color of environmental noise, Ecology, 85 (2004), 1146-1152. Google Scholar

V. Volterra, Variazioni e Fluttuazioni Del Numero D'individui in Specie Animali Conviventi, C. Ferrari, 1927. Google Scholar

X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations of population dynamics, Journal of Mathematical Analysis and Applications, 304 (2005), 296-320. doi: 10.1016/j.jmaa.2004.09.027. Google Scholar

X. Zhao and Z. Zeng, Stationary distribution and extinction of a stochastic ratio-dependent predator–prey system with stage structure for the predator, Physica A: Statistical Mechanics and its Applications, 545 (2020), 123310, 17 pp. doi: 10.1016/j.physa.2019.123310. Google Scholar

##### References:

J. Alebraheem and Y. A. Hasan, Dynamics of a two predator–one prey system, Computational and Applied Mathematics, 33 (2014), 767-780. doi: 10.1007/s40314-013-0093-8. Google Scholar

A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 292 (2004), 364-380. doi: 10.1016/j.jmaa.2003.12.004. Google Scholar

Y. Bai and Y. Li, Stability and Hopf bifurcation for a stage-structured predator–prey model incorporating refuge for prey and additional food for predator, Advances in Difference Equations, 2019 (2019), 1-20. doi: 10.1186/s13662-019-1979-6. Google Scholar

J. Bao and C. Yuan, Stochastic population dynamics driven by Lévy noise, Journal of Mathematical Analysis and Applications, 391 (2012), 363-375. doi: 10.1016/j.jmaa.2012.02.043. Google Scholar

G. A. Bocharov and F. A. Rihan, Numerical modelling in biosciences using delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 183-199. doi: 10.1016/S0377-0427(00)00468-4. Google Scholar

E. Buckwar, Introduction to the numerical analysis of stochastic delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 297-307. doi: 10.1016/S0377-0427(00)00475-1. Google Scholar

B. Dubey and A. Kumar, Dynamics of prey–predator model with stage structure in prey including maturation and gestation delays, Nonlinear Dynamics, 96 (2019), 2653-2679. doi: 10.1007/s11071-019-04951-5. Google Scholar

M. F. Elettreby, Two-prey one-predator model, Chaos, Solitons & Fractals, 39 (2009), 2018-2027. doi: 10.1016/j.chaos.2007.06.058. Google Scholar

T. C. Gard, Persistence in stochastic food web models, Bulletin of Mathematical Biology, 46 (1984), 357-370. doi: 10.1016/S0092-8240(84)80044-0. Google Scholar

D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Review, 43 (2001), 525-546. doi: 10.1137/S0036144500378302. Google Scholar

R. Khasminskii, Stochastic Stability of Differential Equations, Springer-Verlag Berlin Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. Google Scholar

P. E. Kloeden and T. Shardlow, The Milstein scheme for stochastic delay differential equations without using anticipative calculus, Stochastic Analysis and Applications, 30 (2012), 181-202. doi: 10.1080/07362994.2012.628907. Google Scholar

S. Kundu and S. Maitra, Dynamical behaviour of a delayed three species predator–prey model with cooperation among the prey species, Nonlinear Dynamics, 92 (2018), 627-643. doi: 10.1007/s11071-018-4079-3. Google Scholar

D. Li, S. Liu and J. Cui, Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching, Journal of Differential Equations, 263 (2017), 8873-8915. doi: 10.1016/j.jde.2017.08.066. Google Scholar

R. S. Liptser, A strong law of large numbers for local martingales, Stochastics, 3 (1980), 217-228. doi: 10.1080/17442508008833146. Google Scholar

Z. Liu and R. Tan, Impulsive harvesting and stocking in a monod–haldane functional response predator–prey system, Chaos, Solitons & Fractals, 34 (2007), 454-464. doi: 10.1016/j.chaos.2006.03.054. Google Scholar

Q. Liu, D. Jiang, T. Hayat and A. Alsaedi, Stationary distribution and extinction of a stochastic predator–prey model with herd behavior, Journal of the Franklin Institute, 355 (2018), 8177-8193. doi: 10.1016/j.jfranklin.2018.09.013. Google Scholar

A. J. Lotka, Elements of Physical Biology, Baltimore: Williams & Wilkins Co., 1925. Google Scholar

X. Mao, Stochastic Differential Equations and Applications, Elsevier, 2008. doi: 10.1533/9780857099402. Google Scholar

X. Mao, G. Marion and E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Processes and their Applications, 97 (2002), 95-110. doi: 10.1016/S0304-4149(01)00126-0. Google Scholar

X. Mao, S. Sabanis and E. Renshaw, Asymptotic behaviour of the stochastic Lotka-Volterra model, Journal of Mathematical Analysis and Applications, 287 (2003), 141-156. doi: 10.1016/S0022-247X(03)00539-0. Google Scholar

J. D. Murray, Mathematical Biology, Springer New york, 1993. doi: 10.1007/b98869. Google Scholar

R. Rakkiyappan, A. Chandrasekar, F. A. Rihan and S. Lakshmanan, Exponential state estimation of Markovian jumping genetic regulatory networks with mode-dependent probabilistic time-varying delays, Mathematical Biosciences, 25 (2014), 30-53. doi: 10.1016/j.mbs.2014.02.008. Google Scholar

R Rakkiyappan, G. Velmurugan, F. A. Rihan and and S. Lakshmanan, Stability analysis of memristor-based complex-valued recurrent neural networks with time delays, Complexity, 21 (2015), 14-39. doi: 10.1002/cplx.21618. Google Scholar

F. A. Rihan, S. Lakshmanan, A. H. Hashish, R. Rakkiyappan and E. Ahmed, Fractional-order delayed predator–prey systems with functional response, Nonlinear Dynamics, 80 (2015), 777-789. doi: 10.1007/s11071-015-1905-8. Google Scholar

F. A. Rihan, H. J. Alsakaji and C. Rajivganthi, Stability and Hopf bifurcation of three-species prey-predator system with time delays and Allee effect, Complexity, 2020 (2020), 7306412. doi: 10.1155/2020/7306412. Google Scholar

F. A. Rihan, A. A. Azamov and H. J. Al-Sakaji, An inverse problem for delay differential equations: Parameter estimation, nonlinearity, sensitivity, Applied Mathematics & Information Sciences, 12 (2018), 63-74. doi: 10.18576/amis/120106. Google Scholar

F. A. Rihan and H. J. Alsakaji, Persistence and extinction for stochastic delay differential model of prey-predator system with hunting cooperation in predators, Advances in Difference Equations, 124 (2020), 1-22. doi: 10.1186/s13662-020-02579-z. Google Scholar

F.A. Rihan, H.J. Alsakaji and C. Rajivganthi, Stochastic SIRC epidemic model with time-delay for COVID-19, Adv. Differ. Equ., 2020 (2020), 1-20. doi: 10.1186/s13662-020-02964-8. Google Scholar

F. A. Rihan, C. Rajivganthi and P. Muthukumar, Fractional stochastic differential equations with Hilfer fractional derivative: Poisson jumps and optimal control, Discrete Dynamics in Nature and Society, 2017 (2017), Art. ID 5394528, 11 pp. doi: 10.1155/2017/5394528. Google Scholar

T. Saha and M. Bandyopadhyay, Dynamical analysis of a delayed ratio-dependent prey–predator model within fluctuating environment, Applied Mathematics and Computation, 196 (2008), 458-478. doi: 10.1016/j.amc.2007.06.017. Google Scholar

G. Tang, S. Tang and R. A. Cheke, Global analysis of a holling type II predator–prey model with a constant prey refuge, Nonlinear Dynamics, 76 (2014), 635-647. doi: 10.1007/s11071-013-1157-4. Google Scholar

D. A. Vasseuri and P. Yodzis, The color of environmental noise, Ecology, 85 (2004), 1146-1152. Google Scholar

V. Volterra, Variazioni e Fluttuazioni Del Numero D'individui in Specie Animali Conviventi, C. Ferrari, 1927. Google Scholar

X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations of population dynamics, Journal of Mathematical Analysis and Applications, 304 (2005), 296-320. doi: 10.1016/j.jmaa.2004.09.027. Google Scholar

X. Zhao and Z. Zeng, Stationary distribution and extinction of a stochastic ratio-dependent predator–prey system with stage structure for the predator, Physica A: Statistical Mechanics and its Applications, 545 (2020), 123310, 17 pp. doi: 10.1016/j.physa.2019.123310. Google Scholar     Parameters Description \$ r_1 \$ , \$ r_2 \$ intrinsic growth rate for x and y \$ k_1 \$ , \$ k_2 \$ carrying capacity for x and y \$ alpha_1 \$ , \$ alpha_2 \$ rate of predation of preys x and y \$ eta \$ rate of cooperation of preys x and y against predator z \$ delta \$ Predator death rate \$ alpha_3 \$ rate of intra-species competition within the predators \$ a_1 \$ , \$ a_2 \$ transformation rate of predator to preys \$ x \$ and \$ y \$ .
 Parameters Description \$ r_1 \$ , \$ r_2 \$ intrinsic growth rate for x and y \$ k_1 \$ , \$ k_2 \$ carrying capacity for x and y \$ alpha_1 \$ , \$ alpha_2 \$ rate of predation of preys x and y \$ eta \$ rate of cooperation of preys x and y against predator z \$ delta \$ Predator death rate \$ alpha_3 \$ rate of intra-species competition within the predators \$ a_1 \$ , \$ a_2 \$ transformation rate of predator to preys \$ x \$ and \$ y \$ .

Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507

Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124

Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737

Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244

Chengjun Guo, Chengxian Guo, Sameed Ahmed, Xinfeng Liu. Moment stability for nonlinear stochastic growth kinetics of breast cancer stem cells with time-delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2473-2489. doi: 10.3934/dcdsb.2016056

Kexin Wang. Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1699-1714. doi: 10.3934/dcdsb.2019247

Yanan Zhao, Yuguo Lin, Daqing Jiang, Xuerong Mao, Yong Li. Stationary distribution of stochastic SIRS epidemic model with standard incidence. Discrete & Continuous Dynamical Systems - B, 2016, 21 (7) : 2363-2378. doi: 10.3934/dcdsb.2016051

Meng Liu, Chuanzhi Bai, Yi Jin. Population dynamical behavior of a two-predator one-prey stochastic model with time delay. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2513-2538. doi: 10.3934/dcds.2017108

Meng Zhao, Wan-Tong Li, Jia-Feng Cao. A prey-predator model with a free boundary and sign-changing coefficient in time-periodic environment. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3295-3316. doi: 10.3934/dcdsb.2017138

R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423

Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2109-2120. doi: 10.3934/dcdss.2020180

Gianni Gilioli, Sara Pasquali, Fabrizio Ruggeri. Nonlinear functional response parameter estimation in a stochastic predator-prey model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 75-96. doi: 10.3934/mbe.2012.9.75

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Amelia G. Nobile. A non-autonomous stochastic predator-prey model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 167-188. doi: 10.3934/mbe.2014.11.167

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

Xinfu Chen, Yuanwei Qi, Mingxin Wang. Steady states of a strongly coupled prey-predator model. Conference Publications, 2005, 2005 (Special) : 173-180. doi: 10.3934/proc.2005.2005.173

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536

Mingxin Wang, Peter Y. H. Pang. Qualitative analysis of a diffusive variable-territory prey-predator model. Discrete & Continuous Dynamical Systems, 2009, 23 (3) : 1061-1072. doi: 10.3934/dcds.2009.23.1061

Rui Xu, M.A.J. Chaplain, F.A. Davidson. Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 823-831. doi: 10.3934/dcdsb.2004.4.823

Guanqi Liu, Yuwen Wang. Stochastic spatiotemporal diffusive predator-prey systems. Communications on Pure & Applied Analysis, 2018, 17 (1) : 67-84. doi: 10.3934/cpaa.2018005

## Enzymatic reactions

Enzymes are molecules that mediate chemical reactions. A certain set of reactants can be converted into a certain set of products with some energy expenditure. The enzyme reduces the energy threshold for the reaction to occur. The simplest enzymatic reaction involves the conversion of a substrate \$S\$ into a product \$P\$. Enzymes \$E\$ react with the substrate to generate an intermediate product \$I\$. The intermediate product \$I\$ then liberates the enzyme \$E\$ to yield the final product \$P\$. A simple model of this process is given by the following set of reactions

Reaction \$R_1\$ models the combination of a substrate molecule \$S\$ and an enzyme \$E\$ to yield an intermediate specimen \$I\$. The latter may disassociate from the enzyme yielding either the original substrate \$S\$ (Reaction \$R_2\$) or the the final product \$P\$ (Reaction \$R_3\$). Hazards can be determined using the ideas in previous examples. It is ready to realize that hazards can be written as \$h_1(S,E,I,P)=c_1 S E\$, \$h_2(S,E,I,P)=c_2 I\$ and \$h_3(S,E,I,P)=c_3 I\$. Deterministic simulation of enzymatic reactions is in the file ssas_enzyme_deterministic. Stochastic simulation of enzymatic reactions is in ssas_enzyme_stochastic.m.

## Introduction

In the past, unexpected large quotas in animal and fish catches had been occasionally reported. The phenomenon was attributed to the predators in those communities 1,2,3,4 . This predator and prey relationship is probably the most studied ecological dynamics in recent history. Theoretical studies of this system began when Alfred Lotka and Vito Volterra independently developed the well-known predator-prey model in the 1920s. Lotka developed the model to study autocatalytic chemical reactions and Volterra extended it to explain the fish catches in the Adriatic Sea. Since the earliest developments of the basic Lotka-Volterra system (LV system) 5,6,7,8,9,10 , many mathematical variations of predator-prey systems have been developed to explain unexpected changes and temporal fluctuations in the dynamics of animal populations. This system is considered to explain the dynamics of natural populations of snowshoe hare (Lepus americanus, the prey) and Canadian lynx (Lynx canadensis, the predator) that were estimated from the yearly changes in the collected number of furs 11 . In addition, unlike in natural communities, it has been shown that long-term coexistence of predators and prey is possible in a laboratory using predatory and prey mites 12 . Thus, the LV system has become the classical mathematical model for explaining the predator-prey interactions in natural communities 5,6,7,8,9,10,11 .

Mathematical properties of the classical LV system show either cyclic oscillation or divergent extinction of one species 13 . In any closed LV system, it is also important to note that the predators will eventually die out with the extinction of the preys. This means that the persistent predator-prey systems, without additional stabilizing mechanisms, should exhibit cyclic oscillations. However, stable coexistence in wild predator-prey systems has been observed 10 . Nonetheless, the case of the lynx-hare interaction 8 seems extremely unique natural predator-prey system. We often find stable coexistence of both prey and predator populations in the wild such as that of spider wasps (Pompilidae family, the predator) and spiders (the prey) 10 . These observations imply that there should be an additional stabilizing mechanism in natural predator-prey systems. For example, strong intraspecific competition in both predator and prey yields stable coexistence 14 . However, we have no evidence of such strong intraspecific competition in the wild. Thus, these observations of the most natural systems contradict with the original solution of the classical LV system.

In this paper, we explore the convergent solutions in predator-prey systems by modifying the classical LV system. Because most predator-prey systems in the wild are not isolated, we consider the effects of fixed (or random) number of immigrants at regular intervals on the predator and prey populations. By adding few immigrants, the LV systems with type I, II, and III functional responses exhibit asymptotic stability. Similarly, adding few immigrants to the predator population stabilizes the modified LV systems where both predator and prey coexist. In the latter case, the LV system may be interpreted as a host-parasite system, because the parasites are more likely to become immigrants. We then briefly discuss the implications of the modified LV system on the predator-prey systems found in the wild.

## On stochastic Lotka-Volterra predator-prey model - Biology

p-ISSN: 2325-0046 e-ISSN: 2325-0054

### Utility Functions and Lotka-Volterra Model: A Possible Connection in Predator-Prey Game

Institute of Radiology, Faculty of Medicine and Surgery, Second University of Naples, Naples, Italy

Correspondence to: Nicola Serra, Institute of Radiology, Faculty of Medicine and Surgery, Second University of Naples, Naples, Italy.

 Email:

The present paper deals a preliminary study on possible connection between the Lotka-Volterra model and predator-prey utility functions. Therefore a generalization of the utility functions to predator-prey population is defined, considering that the utility functions depend by parameters as the strategies adopted, physical efficiency of the predator versus the prey, environmental conditions, prey prudence, etc.

Keywords: Prey – predator interaction, Lotka-Volterra model, Utility function, Game theory