import numpy as np

def invasion_fitness(z, zm, pars):
    n_res = z / pars
    return zm - pars * n_res


def invasion_fitness2(z, zm, pars):
    alpha, beta = pars
    n_res = (z - z**beta) / alpha
    return zm - zm**beta - alpha * n_res


def pop_dynamics2(t, n, pars):
    alpha, beta, z = pars
    return n * (z - z**beta - alpha * n)

def invasion_fitness3(z, zm, pars):
    z0, k = pars
    r = np.exp(-((z - z0) ** 2) / 2)
    n_res = r
    rm = np.exp(-((zm - z0) ** 2) / 2)
    am = np.exp(-((zm - z) ** 2) / k)
    return rm - am * n_res