scholarly journals Features of the Computational Implementation of the Algorithm for Estimating the Lyapunov Exponents of Systems with Delay

2019 ◽  
Vol 26 (4) ◽  
pp. 572-582
Author(s):  
Vladimir E. Goryunov
Keyword(s):  
Lyapunov Exponents ◽  
Delay Differential Equations ◽  
Logistic Equation ◽  
The Other ◽  
Numerical Estimation ◽  
Trigonometric Functions ◽  
Computational Implementation ◽  
Delay Differential ◽  
Usage Flexibility ◽  
Stable Structures

We consider the computational implementation of the algorithm for Lyapunov exponents spectrum numerical estimation for delay differential equations. It is known that for such systems, as well as for boundary value problems, it is not possible to prove the well-known Oseledets theorem which allows us to calculate the required parameters very efficiently. Therefore, we can only talk about the estimates of the characteristics in some sense close to the Lyapunov exponents. In this paper, we propose two methods of linearized systems solutions processing. One of them is based on a set of impulse functions, and the other is based on a set of trigonometric functions. We show the usage flexibility of these algorithms in the case of quasi-stable structures when several Lyapunov exponents are close to zero. The developed methods are tested on a logistic equation with a delay, and these tests illustrate the “proximity” of the obtained numerical characteristics and Lyapunov exponents.

scholarly journals Small-scale spatial structure in plankton distributions

2007 ◽  
Vol 4 (2) ◽  
pp. 173-179 ◽  
Author(s):  
A. Tzella ◽  
P. H. Haynes
Keyword(s):  
Differential Equations ◽  
Spatial Structure ◽  
Delay Differential Equations ◽  
Spatial Scales ◽  
Time History ◽  
The Other ◽  
Small Scale ◽  
Maturation Time ◽  
Length Scales ◽  
Delay Differential

Abstract. The observed filamental nature of plankton populations suggests that stirring plays an important role in determining their spatial structure. If diffusive mixing is neglected, the various interacting biological species within a fluid parcel are determined by the parcel time history. The induced spatial structure has been shown to be a result of competition between the time evolution of the biological processes involved and the stirring induced by the flow as measured, for example, by the rate of divergence of the distance of neighbouring fluid parcels. In the work presented here we examine a simple biological model based on delay-differential equations, previously seen in Abraham (1998), including nutrients, phytoplankton and zooplankton, coupled to a strain flow. Previous theoretical investigations made on a differential equation model (Hernández-Garcia et al., 2002) imply that the latter two should share the same small-scale structure. The generalisation from differential equations to delay-differential equations, associated with the addition of a maturation time to the zooplankton growth, should not make a difference, provided sufficiently small spatial scales are considered. However, this theoretical prediction is in contradiction with the results of Abraham (1998), where the phytoplankton and zooplankton structures remain uncorrelated at all length scales. A new set of numerical experiments is performed here which show that these two regimes coexist. On larger scales, there is a decoupling of the spatial structure of the zooplankton distribution on the one hand, and the phytoplankton and nutrient on the other. On the other hand, at small enough length scales, the phytoplankton and zooplankton share the same spatial structure as expected by the theory involving no maturation time.

Intra-specific competition and insect larval development: a model with time-dependent delay

2017 ◽  
Vol 147 (2) ◽  
pp. 353-369 ◽  
Author(s):  
Stephen A. Gourley ◽  
Rongsong Liu ◽  
Yijun Lou
Keyword(s):  
Time Delay ◽  
Delay Differential Equations ◽  
The Other ◽  
Time Dependent ◽  
Threshold Condition ◽  
Structured Model ◽  
Larval Phase ◽  
State Dependent ◽  
Independent Variable ◽  
Delay Differential

We derive a stage-structured model for an insect population in which a larva matures on reaching a certain size, and in which there is intra-specific competition among larvae that hinders their development, thereby prolonging the larval phase. The model, a system of delay differential equations for the total numbers of adults and larvae, assumes two forms. One of these is a system with a variable state-dependent time delay determined by a threshold condition, the other has constant and distributed delays, a size-like independent variable replacing time t, and no threshold condition. We prove theorems on boundedness and on the linear stability of equilibria.

Delay-Differential Equation Models for Fisheries

1973 ◽  
Vol 30 (7) ◽  
pp. 939-945 ◽  
Author(s):  
Gilbert G. Walter
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Delay Differential Equations ◽  
Growth Curves ◽  
Fishing Effort ◽  
Oscillatory Behavior ◽  
The Other ◽  
Differential Equation Models ◽  
New Models ◽  
Delay Differential

Two new "simple" fishery models based on delay-differential equations are introduced and compared to three currently used differential equation models. These new models can account for reproductive lag and allow oscillatory behavior of population biomass, but require only catch and effort data for their application. Equilibrium levels are calculated for both models and examples of various types of growth curves are given. Levels of fishing effort which maximize yield are calculated and found in one case to depend on the previous population and in the other to be constant.

NUMERICAL INVESTIGATION OF D-BIFURCATIONS FOR A STOCHASTIC DELAY LOGISTIC EQUATION

2005 ◽  
Vol 05 (02) ◽  
pp. 211-222 ◽  
Author(s):  
NEVILLE J. FORD ◽  
STEWART J. NORTON
Keyword(s):  
Differential Equations ◽  
Lyapunov Exponents ◽  
Delay Differential Equations ◽  
Probability Distributions ◽  
Dynamical Behaviour ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Delay Differential ◽  
Delay Logistic Equation ◽  
Parameter Dependent

This paper explores the use of numerical (approximation) methods in the detection of changes in the dynamical behaviour of solutions to parameter-dependent stochastic delay differential equations. We focus on the use of approximations to Lyapunov exponents. Using three numerical methods we begin to describe the probability distributions of the local approximate Lyapunov exponents and we use this information to enable us to predict values of the parameters at which solutions bifurcate. We conclude the paper by reviewing some of the potential pitfalls of using numerical simulations to detect the dynamical behaviour of the solutions to stochastic delay differential equations.

scholarly journals Small-scale spatial structure in plankton distributions

2006 ◽  
Vol 3 (6) ◽  
pp. 1791-1808 ◽  
Author(s):  
A. Tzella ◽  
P. H. Haynes
Keyword(s):  
Differential Equations ◽  
Spatial Structure ◽  
Delay Differential Equations ◽  
Spatial Scales ◽  
Time History ◽  
The Other ◽  
Small Scale ◽  
Maturation Time ◽  
Length Scales ◽  
Delay Differential

Abstract. The observed filamental nature of plankton populations suggests that stirring plays an important role in determining their spatial structure. If diffusive mixing is neglected, the various interacting biological species within a fluid parcel are determined by the parcel time history. The induced spatial structure has been shown to be a result of competition between the time evolution of the biological processes involved and the stirring induced by the flow as measured, for example, by the rate of divergence of the distance of neighbouring fluid parcels. In the work presented here we examine a simple biological model based on delay-differential equations, previously seen in Abraham (1998) including nutrients, phytoplankton and zooplankton, coupled to a strain flow. Previous theoretical investigations made on a differential equation model (Hernández-Garcia et al., 2002) imply that the latter two should share the same small-scale structure. The generalization from differential equations to delay-differential equations, associated with the addition of a maturation time to the zooplankton growth, should not make a difference, provided sufficiently small spatial scales are considered. However, this theoretical prediction is in contradiction with the results of Abraham (1998) where the phytoplankton and zooplankton structures remain uncorrelated at all length scales. A new set of numerical experiments is performed here which show that these two regimes coexist. On larger scales , there is a decoupling of the spatial structure of the zooplankton distribution on the one hand, and the phytoplankton and nutrient on the other. On the other hand, at small enough length scales, the phytoplankton and zooplankton share the same spatial structure as expected by the theory involving no maturation time.

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