^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

An analysis of dynamics of demand-inventory model with stock-level-dependent demand formulated as a three-dimensional system of difference equations with four parameters is considered. By reducing the model to the planar system with five parameters, an analysis of one-parameter bifurcation of equilibrium points is presented. By the analytical method, we prove that nondegeneracy conditions for the existence of Neimark-Sacker bifurcation for the planar system are fulfilled. To check the sign of the first Lyapunov coefficient of Neimark-Sacker bifurcation, we use numerical simulations. We give phase portraits of the planar system to confirm the previous analytical results and show new interesting complex dynamical behaviours emerging in it. Finally, the economical interpretation of the system is given.

Great economic development after the second world war has released a need of development of mathematical methods to support optimization of economic and business processes. Material flow, production, and inventory are aspects of a business, which, to make it profitable, need to be optimized. Therefore, many models of supply chain were created in the mid-20th century. To mention the most noticeable, we ought to list works of Wagner and Whitin [

Prediction of future demand and inventory is an important aspect of running and managing manufacturing or trade company. Methods supporting those tasks have been developed by economists already in the mid-20th century; nonetheless, they are still being improved, as economy still changes and creates new challenges. One of those methods is modelling of economic phenomena using mathematical formulations. The topic of demand and inventory needs a contextual approach, since many factors can influence it and different views may be needed. Therefore, the models are created with the usage of different mathematical tools. We can mention here recent works related to demand and inventory that investigate and describe specific economic cases: Chen and Hu in [

Ma and Feng in [

The model describes demand and inventory of a product at one echelon of supply chain at the retailer. The considered supply chain consists of three echelons: manufacturer, retailer, and customers. The following rules are applied to the model: customers buy a good from a retailer; a retailer orders a product in the forecasted amount and forecast is prepared using single exponential smoothing Brown model [

The model takes a form of the following system of difference equations:

The first equation describes the influence of relation between current stock

We analyse properties of a given model (

For the sake of convenience, let us write the model as a system of three first-order difference equations with

Let us shortly elaborate on the meaning of those equilibrium points. The goal of the retailer is to reach target stock

We recall that

Here,

One can observe that the second and the third equation of system (

System (

From (

Let us later write our systems (

Let

Let

Mapping (

Stability of the equilibrium points of (

Overview of conditions for existence and stability of the equilibrium points of (

Conditions on parameters | | | |
---|---|---|---|

C1 | | Stable | Does not exist |

C2 | | Unstable | Does not exist |

C3 | | Unstable | Stable |

C4 | | Unstable | Stable |

C5 | | Unstable | Unstable |

Graphs illustrating the cases in Table

Examples illustrating conditions (C1) and (C2) from Table

Example illustrating condition (C2) from Table

Examples illustrating conditions (C3) and (C4) from Table

Examples illustrating condition (C5) from Table

The planar system is dependent on five parameters

From the conditions in Table

For

For

Suppose that a two-dimensional discrete-time system

Then, there are smooth invertible coordinate and parameter changes transforming the system into

For any generic two-dimensional one-parameter system

The genericity conditions assumed in the theorem are the transversality condition (C.1) and the nondegeneracy condition (C.2) from Lemma

(C.3)

We start with translation mapping

In the analogous way, we can prove the existence of the Neimark-Sacker one-parameter bifurcation of the planar system for any other parameters

In this section, by using numeral simulation, we give the bifurcation diagrams and phase portraits of system (

Now, let us give illustration to the cases provided in Table

Interesting behaviour can be observed at border point

Let us have again a look at the border case of

Dynamics of the case

System (

Diagram of bifurcation of the equilibrium points with respect to

Diagram of bifurcation of the equilibrium points with respect to

Parameter

Diagrams of bifurcation of the equilibrium points with respect to

Diagrams of bifurcation of the equilibrium points with respect to

Diagrams of bifurcation of the equilibrium points with respect to

Diagram of bifurcation of the equilibrium points with respect to

Numerical analysis and graph plotting have been performed using Matcontm (according to [

The authors declare that there are no competing interests regarding the publication of this paper.