Problem with data type declaration in a function in Julia

I am currently trying to write some code to simulate a Lotka-Volterra model about a population of preyes and predators.

I have written this code:

function PP_gen(N_prey::Int128, N_pred::Int128, max_time::Int128, λ_breed::Float64, λ_interaction::Float64, λ_death::Float64)::Vector{Int128}
    Prey_list = Vector{Int128}([N_prey])
    Pred_list = Vector{Int128}([N_pred])
    
    t = 1
    
    while t < max_time 
        t += 1
        new_Prey = Int128(0)
        new_Pred = Int128(0)
        
        breed = Int128(rand(Binomial(N_prey, λ_breed)));
        new_Prey = breed
        
        interaction = Int128(rand(Binomial(N_pred, λ_interaction)));
        new_Pred = interaction
        new_Prey -= interaction
        
        death = Int128(rand(Binomial(N_pred, λ_death)));
        new_Pred -= death
        
        N_prey += new_Prey
        N_pred += new_Pred
        
        if N_prey <= 0
            push!(Prey_list, 0);
            push!(Pred_list, N_pred)
            return Prey_list, Pred_list
        end
        
        if N_pred <= 0
            push!(Pred_list, 0);
            push!(Prey_list, N_prey)
            return Prey_list, Pred_list
        end
        
        push!(Prey_list, N_prey)
        push!(Pred_list, N_pred)
    end
    return Prey_list, Pred_list
end

I need to try a high number of iteration (for example 1000) but I can only manage to obtain the variables Prey_list and Pred_list as Vector{Int64}, which is a problem since Int64 incurs in stack overflow too soon.
How can I manage to obtain an object of the type Vector{Int128}?

Changing the top of your method to:

function PP_gen(N_prey::Integer, N_pred::Integer, max_time::Integer, λ_breed::Float64, λ_interaction::Float64, λ_death::Float64)::Vector{Int128}
    Prey_list = Int128[N_prey]
    Pred_list = Int128[N_pred]

should be enough to make your code work.

(if it does not work then as @DFN commented seeing the whole code that is problematic would be useful)