Lanchester's laws
: 7-5–7-8  Lanchester's laws have also been used to model guerrilla warfare.
Attempts have been made to apply Lanchester's laws to conflicts between animal groups. Examples include tests with chimpanzees and ants. The chimpanzee application was relatively successful. A study of Australian meat ants and Argentine ants confirmed the square law, a study of fire ants did not confirm the square law.

Helmbold Parameters: The Helmbold Parameters provide quick, concise, exact numerical indices, soundly based on historical data, for comparing battles with respect to their bitterness and the degree to which side had the advantage. While their definition is modeled after a solution of the Lanchester Square Law's differential equations, their numerical values are based entirely on the initial and final strengths of the opponents and in no way depend upon the validity of Lanchester's Square Law as a model of attrition during the course of a battle.
The solution of Lanchester's Square Law used here can be written as:
  
    
      
        
          
            
              
                a
                (
                t
                )
              
              
                
                =
                cosh
                ⁡
                (
                λ
                t
                )
                −
                μ
                sinh
                ⁡
                (
                λ
                t
                )
              
            
            
              
                d
                (
                t
                )
              
              
                
                =
                cosh
                ⁡
                (
                λ
                t
                )
                −
                
                  μ
                  
                    −
                    1
                  
                
                sinh
                ⁡
                (
                λ
                t
                )
              
            
            
              
                ε
              
              
                
                =
                λ
                T
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}a(t)&=\cosh(\lambda t)-\mu \sinh(\lambda t)\\d(t)&=\cosh(\lambda t)-\mu ^{-1}\sinh(\lambda t)\\\varepsilon &=\lambda T\end{aligned}}}
  
where 
  
    
      
        t
      
    
    {\displaystyle t}
  
 is the time since the battle began, 
  
    
      
        a
        (
        t
        )
      
    
    {\displaystyle a(t)}
  
 and 
  
    
      
        d
        (
        t
        )
      
    
    {\displaystyle d(t)}
  
 are the surviving fractions of the attacker's and defender's forces at time 
  
    
      
        t
      
    
    {\displaystyle t}
  
, 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 is the Helmbold intensity parameter, 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
 is the Helmbold defender's advantage parameter, 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is the duration of the battle, and 
  
    
      
        ε
      
    
    {\displaystyle \varepsilon }
  
 is the Helmbold bitterness parameter.
If the initial and final strengths of the two sides are known it is possible to solve for the parameters 
  
    
      
        a
        (
        T
        )
      
    
    {\displaystyle a(T)}
  
, 
  
    
      
        d
        (
        T
        )
      
    
    {\displaystyle d(T)}
  
, 
  
    
      
        μ
      
    
    {\displaystyle \mu }
  
, and 
  
    
      
        ε
      
    
    {\displaystyle \varepsilon }
  
. If the battle duration 
  
    
      
        T
      
    
    {\displaystyle T}
  
 is also known, then it is possible to solve for 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
.
If, as is normally the case, 
  
    
      
        ε
      
    
    {\displaystyle \varepsilon }
  
 is small enough that the hyperbolic functions can, without any significant error, be replaced by their series expansion up to terms in the first power of 
  
    
      
        ε
      
    
    {\displaystyle \varepsilon }
  
, and if we adopt the following abbreviations for the casualty fractions
  
    
      
        
          
            
              
                
                  F
                  
                    A
                  
                
              
              
                
                =
                1
                −
                a
                (
                T
                )
              
            
            
              
                
                  F
                  
                    D
                  
                
              
              
                
                =
                1
                −
                d
                (
                T
                )
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}F_{A}&=1-a(T)\\F_{D}&=1-d(T)\end{aligned}}}
  
then the following approximate relations hold:
  
    
      
        
          
            
              
                ε
              
              
                
                =
                
                  
                    
                      F
                      
                        A
                      
                    
                    
                      F
                      
                        D
                      
                    
                  
                
              
            
            
              
                μ
              
              
                
                =
                
                  F
                  
                    A
                  
                
                
                  /
                
                
                  F
                  
                    D
                  
                
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}\varepsilon &={\sqrt {F_{A}F_{D}}}\\\mu &=F_{A}/F_{D}\end{aligned}}}
  
That 
  
    
      
        ε
      
    
    {\displaystyle \varepsilon }
  
 is a kind of "average" (specifically, the geometric mean) of the casualty fractions justifies using it as an index of the bitterness of the battle.
We note here that for statistical work it is better to use the natural logarithms of the Helmbold Parameters. We will call them, in an obvious notation, 
  
    
      
        log
        ⁡
        μ
      
    
    {\displaystyle \log \mu }
  
, 
  
    
      
        log
        ⁡
        ε
      
    
    {\displaystyle \log \varepsilon }
  
, and 
  
    
      
        log
        ⁡
        λ
      
    
    {\displaystyle \log \lambda }
  
.

Major findings: See Helmbold (2021):

The Helmbold parameters 
  
    
      
        log
        ⁡
        ε
      
    
    {\displaystyle \log \varepsilon }
  
 and 
  
    
      
        log
        ⁡
        μ
      
    
    {\displaystyle \log \mu }
  
 are statistically independent, i.e., they measure distinct features of a battle.
The probability that the defender wins, P(Dwins), is related to the defender's advantage parameter via the logistic function, P(Dwins) = 1 / (1 + exp(-z)), with z = -0.1794 + 5.8694 * logmu. This logistic function is almost exactly skew-symmetric about logmu = 0, rising from P(Dwins) = 0.1 at logmu = -0.4, through P(DWins) = 0.5 at logmu = 0, to P(Dwins) = 0.9 at logmu = +0.4. Because the probability of victory depends on the Helmbold advantage parameter rather than the force ratio, it is clear that force ratio is an inferior and untrustworthy predictor of victory in battle.
While the defender's advantage varies widely from one battle to the next, on average it has been practically constant since 1600CE.
Most of the other battle parameters (specifically the initial force strengths, initial force ratios, casualty numbers, casualty exchange ratios, battle durations, and distances advanced by the attacker) have changed so slowly since 1600CE that only the most acute observers would be likely to notice any change over their nominal 50-year military career.
Bitterness (
  
    
      
        log
        ⁡
        ε
      
    
    {\displaystyle \log \varepsilon }
  
), casualty fractions (
  
    
      
        
          F
          
            A
          
        
      
    
    {\displaystyle F_{A}}
  
 and 
  
    
      
        
          F
          
            D
          
        
      
    
    {\displaystyle F_{D}}
  
 in the above notation), and intensity (
  
    
      
        log
        ⁡
        λ
      
    
    {\displaystyle \log \lambda }
  
) also changed slowly before 1939CE. But since then they have followed a startlingly steeper declining  curve.
Some observers have noticed a similar post-WWII decline in casualties at the level of wars instead of battles.

See also: Attrition warfare
Lotka–Volterra equations similar mathematical model for predator-prey dynamics
Maneuver warfare
Petrie multiplier similar mathematical model for sexism
Lewis Fry Richardson
Salvo combat model

References: 

Bibliography: Czarnecki, Joseph. N-Squared Law: An Examination of one of the Mathematical Theories behind the Dreadnought Battleship Naval Weapons of the World
Dupuy, Col T N (1979). Numbers, Predictions and War. Macdonald and Jane's.
Helmbold, Robert L. (14 February 1961a). Lanchester Parameters for Some Battles of the Last Two Hundred Years. CORG Staff Paper CORG-SP-122.
Helmbold, Robert L. (1961b). "Lanchester's Equations, Historical Battles, and War Games". Proceedings of the Eighth Military Operations Research Society Symposium, 18–20 October 1961.
Helmbold, Robert L. (12 May 2021). The Key to Victory: Machine Learning the Lessons of History. ISBN 9781668525289.
Lanchester, Frederick W. (1916).