Developing Yield Prediction model for Grapes under Climatic Scenario Along with Disease Management

Crop development and yield are both influenced by the weather. A generic agro-climatic yield prediction model for grape is created and analytically solved in this research. In the field of mathematical biology, this model is valuable for research scholars, faculty members


INTRODUCTION
Grape is one of the most commercially important crops in the world; it has a fairly good source of minerals like calcium, phosphorus, iron and vitamins like B1 and B2. Moreover, the juice is mild, laxative and acts as stimulant for kidneys. It is one of the most ancient crops known to humans. Grapes vines were originally a temperate fruit crop, which is grown successfully under tropical conditions. Unripe grapes are used to treat sore throats, and dried grapes are used against constipation and thirst. Round, ripe, sweet grapes are used to treat a wide range of health problems including cancer, cholera, smallpox, nausea, eye infections, skin, kidney, and liver diseases.
Climate has a profound influence on vine growth, productivity and quality of fruits. Of the factors contributing to the successful cultivation of grapes, climate ranks first. The weather parameters viz., sunlight, rainfall, and humidity also influence the quality development of the fruits.

Downy Mildew (Plasmopara viticola) is known as one of the most important vineyard diseases in Tamil
Nadu because it has the capability to develop and spread very quickly and cause large crop losses in certain areas according to the weather conditions [1]. Farmers must make decisions about whether or not to spray downy mildew and also how frequently to spray and which agrochemicals to use [2]. A good understanding of the stage is needed in incidence and conditions of congenial for the incidence and development of the disease. The efficacy and mode of action of fungicides help the effective management of any disease, particularly downy mildew.
Some mathematical models are developed to provide short-term and field-scale predictions of DM epidemics resulting from infections caused by P. viticola sporangia in Switzerland, France, Austria, Germany, and Italy [3][4][5][6][7][8][9][10]. These models are developed by using a common database of previous publications. Christopher et al. have reformulated the SIR model with host response to infection load for a plant disease [11]. Daniele et al. [12] have developed the model for temporal dynamics of brown rot spreading in fruit orchards. Jeger et al. [13] have developed a generic modelling framework to understand the dynamics of foliar pathogen and bio-control agent (BCA) populations in order to predict the likelihood of successful bio-control in relation to the mechanisms involved. Abdul Latif has formulated the induced resistance to plant disease using a dynamical system approach [14]. Mario de la Fuente has compared different methods of grapevine yield prediction in the time window between the fruit set and version [15]. Rory Ellis et al. [16]  Most of the previous yield prediction models using secondary data, the model obtained in a particular district based on data, cannot apply to other districts. But, this proposed yield prediction model for grapes is generic for all districts.
According to the literature survey, there are many yield-estimating models that can be used to estimate the yield of wheat, rice, maize, sorghum, sugarcane, etc. However, for grapes, there are no models available for estimation without secondary data. So far, no models have been reported for the estimation exactly of grape yield in Indian terrain. The present study aims at developing an agro-climatic grape yield prediction model for the study area in the Theni district based on current and future climate data. However, to the best of our knowledge, till date no general model and analytical results for the concentration of climate, disease and yield of grape as a function of infection rate, disease incidence, seasonality rate and removal rate of grape yield loss per harvest time. The obtained analytical solution in comparison with the numerical and stability analysis is found to be in satisfactory agreement. In addition, the basic reproduction number for the yield prediction model for grape is obtained.

MATHEMATICAL FORMULATION OF THE PROBLEM
In the development of the yield prediction model, temperature, relative humidity, rainfall, and rainy days etc., are all considered climate domain characteristics. Climate is affected by indirectly for grape yield; disease is affected by directly grape yield. Figure 1 shows the agro-climatic disease grape yield model schematic diagram used to define the situation for the real-life assumption of the theoretical outcome.
The parameters from the domain  is the seasonality rate,  is the disease incidence, is the infection rate and  is the removal rate of yield loss per harvest time. It is considered in the development of the agroclimatic grape yield prediction model using the asymptotic analysis. The basic form of the model is indicated below: The corresponding initial conditions are: where C is the concentration of climate, D is the concentration of disease, Y is the concentration of yield, t is the time in days,  is the infection rate for grape,  is the disease incidence rate for grape,  is the seasonality rate,  is the removal rate of grape yield loss per harvest time, using HPM to find the solution of

Equilibria:
An equilibrium point is a point at which variables of a system remain unchanged over time. An equation (1) - and the system is stable at this equilibrium point. If the system is at stable steady state and is perturbed slightly off the steady state, then the system will return to the steady state. Therefore, small fluctuations in crops will not destroy the equilibrium and it would expect to observe such equilibrium in nature. In this way, the stability typically determines physically viable behavior. It is now determined that the behavior of equations (1)-(3) near the equilibrium point finds the linearization at the equilibrium. Jacobian matrix is needed to assess.

NUMERICAL SOLUTION
The model formulation of the equation is numerically solved to test the accuracy of this analytical method. Eqs.
(1-3) are numerically solved using Matlab software, a programme that may be used to solve initial value problems. A complete MATLAB application for numerical simulation is included in A. The comparison confirmed that the numerical results match visually and tabular analytical results extremely well. For using field-level data during the period 2015-2021 (in Table 2), the seasonality rate, the disease incidence, the infection rate and the removal rate of yield loss per harvest time are obtained and applied in the given analytical result. There is no significant difference in error % between the numerical and analytical results.

VALIDATION RESULT
In this study, we also propose a survey of grapes growing areas for incidence of downy mildew from 2015 to 2021. A total of fifteen vineyards were selected for the collection of disease incidence levels. The observations on the disease incidence were collected twice a week from the selected grapes vineyards. The results of the survey conducted on grapes showed that downy mildew was a major disease than other diseases especially 0-Theni district, Tamil Nadu. The daily weather data are taken on average to form year-wise weather data. The     8 shows the three-dimension space on the concentration of climate for varying effective seasonality rate and infection rate. The concentration of climate is independent of both  and  but is a function of * C where reduces the concentration of climate. Fig. 9, the concentration of disease varies with infection rate and disease incidence for large value of t . In this regime, the concentration of disease increases with increasing infection rate when 10   . In figure 10, the disease incidence  is extremely high, when the concentration of yield asymptotically reaches a constant value regardless of  , but it depends on . It can be concluded that the concentration of yield increases, when the seasonality index and disease incidence slightly decrease.
Analytical expression of climate, disease and yield are compared with simulation results in Table 1. The maximum relative error between numerical simulations with the analytical result for the developed model is obtained 0.2832%. Stability analysis is carried out for the developed model using the parametric Jacobian transformation method. Based on the obtained results of the mathematical tests, the developed yield prediction model (Eq.5-7) is recommended for its use to estimate the grape yield. Further, phase portraits, for both linear and non-linear system can be predicted or analyzed using algebraic method. In figure 11, is easy to see that the globally stable state and the both upper and lower are positive state are stable nodes.

Ethics statement
No specific permits were required for the described field studies because no human or animal subjects were involved in this research.

Consent for publication
All the authors agreed to publish the content.

Competing interests
There were no conflicts of interest in the publication of this content.
The dotted line represent the numerical results and solid line represents the analytical results.