(School of Data Science and Software Engineering,Qingdao University,Qingdao 266000)

Abstract:The aim of this thesis is to build a mathematical model to analyze hornets spread.we conduct a detailed analysis of the distribution and diffusion of the hornets in space,and simultaneously construct a corresponding spread model.What's more,the most important aspect of the model is how to quantify the spread capacity.

Key words:Cellular Automaton;Asian giant hornet;Hornets Spread Model

1 Introduction

1.1 Background

Asian giant hornets are the largest species of hornet in the world,and are known as agricultural pests[1-3].The hornets are predators of European honeybees,invading and destroying their nests,and they also prey on insects.People in the state of Washington report sightings of these hornets to governments,but a great many of people mistake other types of insects for hornets.[4]Based on that,the state must prioritize its limited resources to follow-up with additional investigation.

1.2 Problem Restatement &Analysis

Build a mathematical model to predict the spread of the hornet and give the accuracy.Addressing the spread of hornets,the core is to conduct a detailed analysis of the distribution and diffusion of the hornets in space and simultaneously construct a corresponding spread model.What's more,the most important aspect of the model is how to quantify the spread capacity.

2 Assumptions

To simplify the given problems,we make the following assumptions for our models:

b.The evolutionary of cellular automata is ideal.

c.Suppose hornets only nest underground,thus,the number of hornets in cities is negligible.

d.There are very few hornets in the mountains,so suppose hornets don't go to the mountains.

3 Notations

Related descriptions are shown in Table 1.

table 1 Symbolic Description

4 Prediction of the spread of the hornets

Biological population migration is a group behavior,which exhibit the characteristics of chaos and randomness,so we use cellular automata to simulate the migration behavior of hornets.We simulated the land terrain of the study area and divided the entire range into 16x8 blocks,each of which changes with temperature over time.Here we assume that the biggest factor influencing hornets spread is the temperature of area,so the driving force for cell movements comes largely from the pursuit of an appropriate temperature.

4.1 Cellular Automata

Cellular automata is a discrete model which consists of a regular grid of cells.Each grid is in one of a finite number of states.As a result,it is able to simulate the spatiotemporal evolution of complex systems.The spread of hornets is also limited in time and space.It means that CA can be used to solve this problem.The commonly used cellular automaton model is a four-tuple model which is defined as

Figure 1 Map of study area

where L represent a cellular space.C is a set of cellular states.N represents a collection of all cells in a neighborhood.F indicates a local mapping as the model evolution rule.

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Besides,figure 2 shows the most common types of neighborhoods are the Von-Neumann neighborhood and the Moore neighborhood.

Figure 2 Neighborhood(a) Von Neumann(b) Moore

We use the Moore-type cellular automaton to build the Hornets Spread Model.The neighbors of the Moore-type cellular automaton are 8 cells around the cell,while 4 cells of the Von Neuman-type cell.Set the research area as cell space with a size of 300x300 pixels.Each block has a different temperature.Based on the above assumptions for hornets,in each step,every small area of hornets,that is,a cell,has two modes of action.When hornets are in an appropriate temperature area,they will randomly choose a nearby position and move randomly.The probability depends on the number of nearby hornets.The more of the numbers of nearby hornets,the greater the probability of moving.This is to simulate the behavior of hornets in allocating living resources.

In order to make it better adapt to this problem,the cell state and evolution rules of the CA are improved.The state of traditional CA is 1 or 0,which has the drawback to quantify hornets dissemination.Therefore,in this paper,the state of CA represents the ability of hornets dissemination,thus the evolution rule is that:the number of wasps in Washington is the initial value of cellular automata evolution.

There are several different states of the cell:states where wasp exists are set as 1,mountains state is set as 2,valley rivers state are set as 3,and plains state are set as 4.By default,at the beginning,there are no wasps in mountains,plains,and valley rivers.When wasps spread to mountains,they do not build nests on mountains,so there are no wasps appear in the state,the value is still in mountains state 2.When nesting in plain or valley rivers,the wasps start to appear in the state,that is,4 and 3 become 1;because wasps spread randomly,and the number of warm places in valley rivers is more affluent,Moran's I and weight model are added,that is,compared with plain,the wasps tend to nest in Valley rivers.

4.2 Hornets Spread Model

In order to make calculations easier and more convenient for later modeling,we divide the study area into 16×8 blocks,each block is given a fixed number (Figure3),the longitude and latitude spanned by each block are 0.5° respectively.We assume that the temperature in each block is the same.Take the block's center point temperature as a representative of the temperature of the block,and study 128 blocks separately.Each block is regarded as a point to predict the temperature changes.

Figure 3 16x8 map division

Given the growth trend of hornets’ transmission capacity,which consists of macro and local.The macro means that the spread state of the hornets in space and it named as macro-growth.The local means that their own state with respect to hornets and it's named selfgrowth.As a result,the following expressions are defined to describe two trends.

Considering that self-growth is affected by a large number of factors in real life.To our best knowledge,regardless of how many factors are involved,the overall growth distribution follows a normal distribution.Therefore,in this paper a self-growth model based on normal distribution is proposed to describe self-growth of cells,

According to the principle of CA,the cells in a domain is influenced by the central cell in each evolution process.The effective is positively related to the state of the central cell.Then the macro growth evolution formula of the cells in the domain can be obtained as follows:

However,the parameter k is random in space which satisfies a normal distribution with a mean ofand a variance of σ2.As a result,the macro-growth evolution formula of the cells can be changed to

Where is a parameter to control the external spread velocity.The probability formula is also based on normal distribution and it can be calculated by the formula.

Through our improved Cellular Automata Model,we can get the prediction result chart of hornet transmission.

4.3 Moran's I

In addition,in order to analyze the relationship between spatial density and hornet propagation,we introduce Moran's I into theevolution rule of cellular automata,the Moran's I has the following characteristics.First,Moran's I is normalized to [-1,1].Based on the size of the scope,it can be divided into Global Moran's I and Anselin Local Moran's I,which are used to explain the spatial correlation of an indicator in the region or a local area.Moran's I >0 indicates positive spatial correlation.The larger the value,the stronger the spatial correlation.Moran's I <0 denotes negative spatial correlation.The smaller the value,the larger the spatial difference.Besides,when Moran's I=0,the space is random.

Figure 4 the prediction result chart of hornet transmission

The Global Moran's I can be expressed as

where ω ij is the spatial weight of elements i and j.n indicate the total number of elements.So is the sum of all spatial weights and it can be obtained as

Through Moran's I,we can get the distribution of wasps.The following is the spatial density analysis with latitude and longitude as abscissa and month as ordinate.

Combining with Moran's I,an improved CA model of compound propagation mode is established as follows

Figure 5

By introducing Moran's I into the evolution rules of cellular automata,we can get the prediction result chart of hornet transmission address

Figure 6 Prediction of the spread of hornets

5 The evidence of hornets' eradicated in Washington State

According to the improved cellular automata with Moran’s I index,we deduce the wasps propagation model;we can easily find out that according to the rules of cellular automata defined by us,when the model becomes convergent,that is,the evolution of cellular automata tends to be stable,we can judge that there are no wasps in Washington state.

Figure 7 The population variance chart

Figure 8 Trend forecast chart of propagation quantity

According to the normal distribution of the model,the broken line trend chart of wasp transmission number over time and the population variance chart are given below.It is obvious that in early 2021,the broken line trend chart will be close to the X axis,and the wasp population in Washington will tend to decline and even eradicated.