Lotka-Volterra model:

dR1=R1(1-R1)-a1 R1 N1

dN1=a1 R1 N1-d N1

Beddington model:

dR2=R2(1-R2)-a2 R2 N2/(1+R2/h+N2/h)

dN2=a2 R2 N2/(1+R2/h+N2/h)-d N2

solLV=Solve[{dR1==0,dN1==0},{R1,N1}]

barR1=R1/.solLV[[3]];barN1=N1/.solLV[[3]];

solB=Solve[{dR2==0,dN2==0},{R2,N2}]

barR2=R2/.solB[[4]];barN2=N2/.solB[[4]]

Solve the a2 parameter from the requirement that both states are identical

Solve[{barR1==barR2,barN1==barN2},a2]